Recent zbMATH articles in MSC 65https://zbmath.org/atom/cc/652023-09-19T14:22:37.575876ZUnknown authorWerkzeugBayesian integrals on toric varietieshttps://zbmath.org/1516.140942023-09-19T14:22:37.575876Z"Borinsky, Michael"https://zbmath.org/authors/?q=ai:borinsky.michael"Sattelberger, Anna-Laura"https://zbmath.org/authors/?q=ai:sattelberger.anna-laura"Sturmfels, Bernd"https://zbmath.org/authors/?q=ai:sturmfels.bernd"Telen, Simon"https://zbmath.org/authors/?q=ai:telen.simonThe starting point of this article is the fact that every projective toric variety \(X\) is a positive geometry. With this in mind, the authors use the canonical differential form associated to each variety to represent probability measures over its positive part \(X_{>0}\).
Then the focus is mainly on two tasks:
\begin{itemize}
\item[1.] Evaluating marginal likelihood integrals,
\item[2.] Sampling from the posterior distribution on the positive part of a projective variety.
\end{itemize}
The approach to the tasks is to move to the tropical setting and compute a tropicalization of the marginal likelihood integrals . This approach is effective due to the fact that the likelihood integral can be written as the integral of the product of the canonical form associated to \(X\) and a rational function. In section 3 of the article the authors develop the tropical setting. The linking point between the likelihood integral and its tropical version is given by formula (3.2), and the main results consist on proving that integrals on tropical sectors are finite, and that it is possible to sample from tropical densities.
In section 4 the authors use the results from section 3 to compute numerically the marginal likelihood integrals. Proposition 4.2 guarantees that this approximation is good as its standard deviation stays bounded. At the end of this section the authors also provide an algorithm (Algorithm 2) to sample from the posterior distribution using the sampling process of the tropical densities mentioned in Section 3.
Sections 5 and 6 focus on statistical models and bayesian inference. They showcase the reach of the results from the previous sections and are supported by computations and examples that can be found in MathRepo.
Reviewer: Angelica Torres (Bellaterra)Tolerance and control solutions of two-sided interval linear system and their applicationshttps://zbmath.org/1516.150022023-09-19T14:22:37.575876Z"Leela-apiradee, Worrawate"https://zbmath.org/authors/?q=ai:leela-apiradee.worrawate"Thipwiwatpotjana, Phantipa"https://zbmath.org/authors/?q=ai:thipwiwatpotjana.phantipa"Gorka, Artur"https://zbmath.org/authors/?q=ai:gorka.arturSummary: This work investigates tolerance and control solutions to a two-sided interval linear system. Their semantics are different, even though, we would be able to interchange the role of the interval information algebraically. We present necessary and sufficient conditions of their solvabilities as the inequalities depending on center and radius of coefficient interval matrices on both sides of the system. In a situation when the vector of variables is nonnegative, the conditions can simply be modified as the inequalities depending on boundaries of the interval matrices. This result helps to find out the feasible solutions of a quadratic programming problem with two-sided interval linear equation constraints.
For the entire collection see [Zbl 1481.68017].Computing the square root of a low-rank perturbation of the scaled identity matrixhttps://zbmath.org/1516.150042023-09-19T14:22:37.575876Z"Fasi, Massimiliano"https://zbmath.org/authors/?q=ai:fasi.massimiliano"Higham, Nicholas J."https://zbmath.org/authors/?q=ai:higham.nicholas-j"Liu, Xiaobo"https://zbmath.org/authors/?q=ai:liu.xiaobo.4|liu.xiaobo.2|liu.xiaobo.1|liu.xiaobo.3A \(p\)-th root of a square matrix \(A\) is any solution of the nonlinear equation \(X^p=A\).
Here the authors study methods for computing the square root of a matrix \(A\in\mathbb{C}^{n\times n}\) of the form
\[
A=\alpha I_n+UV^*,~ \alpha\in\mathbb{C},~ U, V\in\mathbb{C}^{n\times k},~ k\le n,~ \Lambda(A)\cap\mathbb{R}^-=\emptyset ,
\]
where \(I_n\) is the identity matrix of order \(n\) and \(\Lambda(A)\) denotes the spectrum of \(A\).
Further, the authors show that for any integer \(p\ge 1\),
\[
A^{1/p}=\alpha^{1/p}I_n+U\left(\sum_{i=0}^{p-1}\alpha^{i/p}\cdot(\alpha I_k+V^*U)^{(p-i-1)/p}\right)^{-1}V^*.
\]
Some applications of the obtained results are discussed.
Reviewer: Ali Morassaei (Zanjan)A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matriceshttps://zbmath.org/1516.150052023-09-19T14:22:37.575876Z"Bolten, Matthias"https://zbmath.org/authors/?q=ai:bolten.matthias"Ekström, Sven-Erik"https://zbmath.org/authors/?q=ai:ekstrom.sven-erik"Furci, Isabella"https://zbmath.org/authors/?q=ai:furci.isabella"Serra-Capizzano, Stefano"https://zbmath.org/authors/?q=ai:serra-capizzano.stefanoSummary: The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.First-order perturbation theory for eigenvalues and eigenvectorshttps://zbmath.org/1516.150062023-09-19T14:22:37.575876Z"Greenbaum, Anne"https://zbmath.org/authors/?q=ai:greenbaum.anne"Li, Ren-Cang"https://zbmath.org/authors/?q=ai:li.rencang"Overton, Michael L."https://zbmath.org/authors/?q=ai:overton.michael-lSummary: We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so well known outside a community of specialists. We give two different proofs of the main eigenvector perturbation theorem. The first, a block-diagonalization technique inspired by the numerical linear algebra research community and based on the implicit function theorem, has apparently not appeared in the literature in this form. The second, based on complex function theory and on eigenprojectors, as is standard in analytic perturbation theory, is a simplified version of well-known results in the literature. The second derivation uses a convenient normalization of the right and left eigenvectors defined in terms of the associated eigenprojector, but although this dates back to the 1950s, it is rarely discussed in the literature. We then show how the eigenvector perturbation theory is easily extended to handle other normalizations that are often used in practice. We also explain how to verify the perturbation results computationally. We conclude with some remarks about difficulties introduced by multiple eigenvalues and give references to work on perturbation of invariant subspaces corresponding to multiple or clustered eigenvalues. Throughout the paper we give extensive bibliographic commentary and references for further reading.Total negativity: characterizations and single-vector testshttps://zbmath.org/1516.150202023-09-19T14:22:37.575876Z"Choudhury, Projesh Nath"https://zbmath.org/authors/?q=ai:choudhury.projesh-nathSummary: A matrix is called totally negative (totally non-positive) of order \(k\), if all its minors of size at most \(k\) are negative (non-positive). The objective of this article is to provide several novel characterizations of total negativity via the (a) variation diminishing property, (b) sign non-reversal property, and (c) Linear Complementarity Problem. (The last of these provides a novel connection between total negativity and optimization/game theory.) More strongly, each of these three characterizations uses a single test vector whose coordinates alternate in sign. As an application of the sign non-reversal property, we study the interval hull of two rectangular matrices. In particular, we identify two matrices \(C^\pm(A, B)\) in the interval hull of matrices \(A\) and \(B\) that test total negativity of order \(k\), simultaneously for the entire interval hull. We also show analogous characterizations for totally non-positive matrices and provide a finite set of test matrices to detect the total non-positivity property of an interval hull. These novel characterizations may be considered similar in spirit to fundamental results characterizing totally positive matrices by \textit{L. D. Brown} et al. [J. Am. Stat. Assoc. 76, 824--832 (1981; Zbl 0481.62021)] (see also [\textit{F. R. Gantmacher} and \textit{M. G. Krein}, Oscillation matrices, oscillation kernels, and small oscillations of mechanical systems (1941; Zbl 0060.03303); \textit{P. N. Choudhury} et al., Bull. Lond. Math. Soc. 53, No. 4, 981--990 (2021; Zbl 1478.15047); \textit{P. N. Choudhury}, Bull. Lond. Math. Soc. 54, No. 2, 791--811 (2022)]). Finally, we show that totally negative/non-positive matrices can not be detected by (single) test vectors from orthants other than the open bi-orthant that have coordinates with alternating signs, via the variation diminishing property or the sign non-reversal property.Łojasiewicz gradient inequalities for polynomial functions and some applicationshttps://zbmath.org/1516.260102023-09-19T14:22:37.575876Z"Ha, Huy Vui"https://zbmath.org/authors/?q=ai:ha-huy-vui."Nguyen, Thi Thao"https://zbmath.org/authors/?q=ai:nguyen.thi-thaoGiven a non-constant polynomial \(f \colon \mathbb{R}^n \to \mathbb{R}\) and a number \(\lambda \in \mathbb{R},\) the authors study the existence of the following Łojasiewicz gradient inequality \[\|\nabla f(x)\| \ge c \min\{|f(x) - \lambda|^{\theta}, |f(x) - \lambda|^{\mu} \} \quad \textrm{ for all } x \in \Omega,\] where either \(\Omega = \mathbb{R}^n\) or \(\Omega\) is a neighborhood of infinity, \(c > 0, \theta\) and \(\mu\) are real numbers. The cases where \(\theta\) and \(\mu\) are nonnegative and belong to \([0, 1)\) are considedred carefully. Some applications in optimization are given.
Reviewer: Tien-Son Pham (Dalat)Existence and uniqueness results for fractional Langevin equations on a star graphhttps://zbmath.org/1516.340232023-09-19T14:22:37.575876Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.26"Zhang, Jifeng"https://zbmath.org/authors/?q=ai:zhang.jifeng"Ni, Jinbo"https://zbmath.org/authors/?q=ai:ni.jinbo(no abstract)Floquet theory of time-varying differential algebraic equations transferable into standard canonical formhttps://zbmath.org/1516.340242023-09-19T14:22:37.575876Z"Younus, Awais"https://zbmath.org/authors/?q=ai:younus.awais"Yasmeen, Nudrat"https://zbmath.org/authors/?q=ai:yasmeen.nudratSummary: This paper deals with time-varying periodic differential algebraic equations (DAEs) transferable into standard canonical form (SCF). The solution theory of linear differential algebraic equations is considered in detail and well known concept of ordinary algebraic equations is generalized for DAEs. We note that Floquet theory of DAEs demands a strict constant rank condition on the singular coefficient matrix which is not required by solution theory of periodic DAEs transferable into SCF. Moreover, we obtain the Floquet type result for periodic DAEs which is transferable into SCF.Analytical and numerical solutions of a class of generalised Lane-Emden equationshttps://zbmath.org/1516.340572023-09-19T14:22:37.575876Z"Awonusika, Richard Olu"https://zbmath.org/authors/?q=ai:awonusika.richard-olu"Oluwafemi Olatunji, Peter"https://zbmath.org/authors/?q=ai:oluwafemi-olatunji.peterSummary: The classical equation of Jonathan Homer Lane and Robert Emden, a nonlinear second-order ordinary differential equation, models the isothermal spherical clouded gases under the influence of the mutual attractive interaction between the gases' molecules. In this paper, the Adomian decomposition method (ADM) is presented to obtain highly accurate and reliable analytical solutions of a class of generalised Lane-Emden equations with strong nonlinearities. The nonlinear term \(f(y(x))\) of the proposed problem is given by the integer powers of a continuous real-valued function \(h(y(x))\), that is, \(f(y(x)) = h^m(y(x))\), for integer \(m\geq 0\), real \(x > 0\). In the end, numerical comparisons are presented between the analytical results obtained using the ADM and numerical solutions using the eighth-order nested second derivative two-step Runge-Kutta method (NSDTSRKM) to illustrate the reliability, accuracy, effectiveness and convenience of the proposed methods. The special cases \(h(y) = \sin y(x), \cos y(x)\); \(h(y) = \sinh y(x), \cosh y(x)\) are considered explicitly using both methods. Interestingly, in each of these methods, a unified result is presented for an integer power of any continuous real-valued function - compared with the case by case computations for the nonlinear functions \(f(y)\). The results presented in this paper are a generalisation of several published results. Several examples are given to illustrate the proposed methods. Tables of expansion coefficients of the series solutions of some special Lane-Emden type equations are presented. Comparisons of the two results indicate that both methods are reliably and accurately efficient in solving a class of singular strongly nonlinear ordinary differential equations.Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillatorhttps://zbmath.org/1516.340602023-09-19T14:22:37.575876Z"Alyousef, Haifa A."https://zbmath.org/authors/?q=ai:alyousef.haifa-a"Alharthi, M. R."https://zbmath.org/authors/?q=ai:alharthi.muteb-r"Salas, Alvaro H."https://zbmath.org/authors/?q=ai:salas.alvaro-h"El-Tantawy, S. A."https://zbmath.org/authors/?q=ai:el-tantawy.s-a(no abstract)On fractal patterns for multi-wing hyperchaotic attractors with a mirror symmetrical structurehttps://zbmath.org/1516.340732023-09-19T14:22:37.575876Z"Doungmo Goufo, Emile F."https://zbmath.org/authors/?q=ai:doungmo-goufo.emile-francSummary: Hyperchaos remains one of the most complex behaviors in the bifurcation mechanisms and so far, only few hyperchaotic dynamics have been identified experimentally. The construction and design of complex models able to generate hyperchaotic attractors with wings on many rows and columns have become a source of interest for many fractal \& chaos theorists, applied physicists and engineers. In this paper, we make use of a simple method, that consists on combining the fractal and fractional operator with Lü system, in order to generate second class hyperchaotic attractors with wings on many rows and columns. Such a combination yields a modified initial value problem, that is solved both analytically and numerically. We then implement the proposed scheme to perform some graphical representations showing the second class types of attractors in the form \(n \times m\)-wings, \((n, m \in \mathbb{N})\), which appear to be hyperchaotic and exhibit a mirror symmetrical structure. The graphical simulations also depict a process where the lower and upper parts of the second class hyperchaotic attractors are seen to be moving away from the mirror symmetrical junction due to the parameter's impact of the fractal-fractional operator.Application of the \(j\)-subgradient in a problem of electropermeabilizationhttps://zbmath.org/1516.350122023-09-19T14:22:37.575876Z"Belhachmi, Zakaria"https://zbmath.org/authors/?q=ai:belhachmi.zakaria"Chill, Ralph"https://zbmath.org/authors/?q=ai:chill.ralphSummary: We study a coupled elliptic-parabolic Poincaré-Steklov system arising in electrical cell activity in biological tissues. By using the notion of \(j\)-subgradient, we show that this system has a gradient structure and thus obtain wellposedness. We further exploit the gradient structure for the discretisation of the problem and provide numerical experiments.Blow-up analysis for a class of higher-order viscoelastic inverse problem with positive initial energy and boundary feedbackhttps://zbmath.org/1516.351252023-09-19T14:22:37.575876Z"Shahrouzi, Mohammad"https://zbmath.org/authors/?q=ai:shahrouzi.mohammadSummary: In this paper, we consider a nonlinear higher-order viscoelastic inverse problem with memory in the boundary. Under some suitable conditions on the coefficients, relaxation function and initial data, we prove a blow-up result for the solution with positive initial energy.Area preserving geodesic curvature driven flow of closed curves on a surfacehttps://zbmath.org/1516.352302023-09-19T14:22:37.575876Z"Kolář, Miroslav"https://zbmath.org/authors/?q=ai:kolar.miroslav"Beneš, Michal"https://zbmath.org/authors/?q=ai:benes.michal"Ševčovič, Daniel"https://zbmath.org/authors/?q=ai:sevcovic.danielSummary: We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.Optimal sensor placement: a robust approachhttps://zbmath.org/1516.352532023-09-19T14:22:37.575876Z"Hintermüller, Michael"https://zbmath.org/authors/?q=ai:hintermuller.michael"Rautenberg, Carlos N."https://zbmath.org/authors/?q=ai:rautenberg.carlos-n"Mohammadi, Masoumeh"https://zbmath.org/authors/?q=ai:mohammadi.masoumeh"Kanitsar, Martin"https://zbmath.org/authors/?q=ai:kanitsar.martinSummary: We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. This paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem, and finalizes with a range of numerical tests.Damped wave systems on networks: exponential stability and uniform approximationshttps://zbmath.org/1516.352622023-09-19T14:22:37.575876Z"Egger, H."https://zbmath.org/authors/?q=ai:egger.hartmut|egger.herbert"Kugler, T."https://zbmath.org/authors/?q=ai:kugler.thomas|kugler.tamarSummary: We consider a damped linear hyperbolic system modeling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a unique steady state is proven in the absence of driving forces. Under mild assumptions on the network topology and the model parameters, we show exponential stability and convergence to equilibrium. This generalizes related results for single pipes and multi-dimensional domains to the network context. Our proofs are based on a variational formulation of the problem, some graph theoretic results, and appropriate energy estimates. These arguments are rather generic and allow us to consider also Galerkin approximations and to prove the uniform exponential stability of the resulting semi-discretizations under mild compatibility conditions on the approximation spaces. A subsequent time discretization by implicit Runge-Kutta methods then allows to obtain fully discrete schemes with uniform exponential decay behavior.
A particular realization by mixed finite elements is discussed and the theoretical results are illustrated by numerical tests in which also bounds for the decay rate are investigated.Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded cornerhttps://zbmath.org/1516.352792023-09-19T14:22:37.575876Z"Chesnel, Lucas"https://zbmath.org/authors/?q=ai:chesnel.lucas"Claeys, Xavier"https://zbmath.org/authors/?q=ai:claeys.xavier"Nazarov, Sergei A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: We investigate the eigenvalue problem \(-\operatorname{div}(\sigma\nabla u) = \lambda u (\mathcal{P})\) in a 2D domain \(\varOmega\) divided into two regions \(\varOmega_{\pm}\). We are interested in situations where \({\sigma}\) takes positive values on \({\varOmega}_{+}\) and negative ones on \({\varOmega}_{-}\). Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [Asymptotic Anal. 88, No. 1--2, 43--74 (2014; Zbl 1294.35153)], we highlighted an unusual instability phenomenon for the source term problem associated with \((\mathcal{P})\): for certain configurations, when the interface between the subdomains \({\varOmega}_{\pm}\) presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem \((\mathcal{P})\). We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.Modeling and numerical simulations of multilane vehicular traffic by active particles methodshttps://zbmath.org/1516.352992023-09-19T14:22:37.575876Z"Zagour, M."https://zbmath.org/authors/?q=ai:zagour.mohamedSummary: This paper deals with the modeling and numerical simulations of multilane vehicular traffic according to the kinetic theory of active particles methods. The main idea of this theory is to consider each driver-vehicle system as a micro-system, where the microscopic state of particles is described by position, velocity, and activity which is an appropriate variable for modeling the quality of the driver-vehicle. The interactions between micro-systems are modeled by stochastic game theory. This leads to the derivation of a mathematical model within the framework of the approach of kinetic theory. The well-posedness of the related Cauchy problem for the spatially homogeneous case is established. Numerical simulations are carried out to show the ability of the proposed model to reproduce the empirical data such as the asymptotic property in time and the emerging behavior of clusters with a particular focus on the road environment conditions.Transport of congestion in two-phase compressible/incompressible flowshttps://zbmath.org/1516.353042023-09-19T14:22:37.575876Z"Degond, Pierre"https://zbmath.org/authors/?q=ai:degond.pierre"Minakowski, Piotr"https://zbmath.org/authors/?q=ai:minakowski.piotr"Zatorska, Ewelina"https://zbmath.org/authors/?q=ai:zatorska.ewelinaSummary: We study the existence of weak solutions to the two-phase fluid model with congestion constraint. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime \(\rho(t, x)\) achieves a threshold value \(\rho^\ast(t, x)\) that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier-Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain.Intrusive and non-intrusive chaos approximation for a two-dimensional steady state Navier-Stokes system with random forcinghttps://zbmath.org/1516.353082023-09-19T14:22:37.575876Z"Lototsky, S. V."https://zbmath.org/authors/?q=ai:lototsky.sergey-v"Mikulevicius, R."https://zbmath.org/authors/?q=ai:mikulevicius.remigijus"Rozovsky, B. L."https://zbmath.org/authors/?q=ai:rozovskii.boris-lSummary: While convergence of a chaos approximation for linear equations is relatively well understood, a lot less is known for non-linear equations. The paper investigates this convergence, by establishing the corresponding a priori error bounds, for a particular equation with quadratic nonlinearity and for two different approximations: stochastic Galerkin and discrete projection. Stochastic Galerkin approximation reduces the stochastic equation to a system of deterministic equation to compute the coefficients in the chaos expansion. The approximation is called intrusive because the resulting system of equations is highly coupled and is harder to solve than the original system; there is also a special condition for uniqueness of solution. An alternative approximation of the chaos coefficients, using the discrete projection version of the stochastic collocation method, is non-intrusive and requires the solution of the original equation for specially chosen realizations of the random input. Compared to the Galerkin approximation, this non-intrusive procedure is easier to analyze and implement, but the resulting approximation error and computational costs can be higher.Gaussian process hydrodynamicshttps://zbmath.org/1516.353102023-09-19T14:22:37.575876Z"Owhadi, H."https://zbmath.org/authors/?q=ai:owhadi.houman|owhadi.howman(no abstract)Kinetic-fluid derivation and mathematical analysis of the cross-diffusion-Brinkman systemhttps://zbmath.org/1516.353252023-09-19T14:22:37.575876Z"Bendahmane, Mostafa"https://zbmath.org/authors/?q=ai:bendahmane.mostafa"Karami, Fahd"https://zbmath.org/authors/?q=ai:karami.fahd"Zagour, Mohamed"https://zbmath.org/authors/?q=ai:zagour.mohamedSummary: In this paper, we propose a new nonlinear model describing the dynamical interaction of two species within a viscous flow. The proposed model is a cross-diffusion system coupled with the Brinkman problem written in terms of velocity fluid, vorticity, and pressure and describing the flow patterns driven by an external source depending on the distribution of species. In the first part, we derive macroscopic models from the kinetic-fluid equations by using the micro-macro decomposition method. On the basis of the Schauder fixed-point theory, we prove the existence of weak solutions for the derived model in the second part. The last part is devoted to developing a one-dimensional finite volume approximation for the kinetic-fluid model, which is uniformly stable along the transition from kinetic to macroscopic regimes. Our computation method is validated with various numerical tests.Complete classification of solutions to the Riemann initial value problem for the Hirota equation with weak dispersion termhttps://zbmath.org/1516.353702023-09-19T14:22:37.575876Z"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.7"Li, Erbo"https://zbmath.org/authors/?q=ai:li.erbo"Xue, Yushan"https://zbmath.org/authors/?q=ai:xue.yushanSummary: In this paper, the Riemann problem for the defocusing Hirota equation with weak dispersion is investigated with Whitham modulation theory. Hirota equation can effectively describe the realistic wave motion in dispersive medium. Via averaging Lagrangian method, the Whitham modulation equations in slow modulation form are obtained, which are characterized by wave parameters and reflects the dispersion relation in the original system. Besides, the modulation equations in Riemann invariant form are derived via finite-gap integration theory. Utilizing Whitham modulation equations parameterized by Riemann invariants, the basic structures of solutions for the Riemann problem for original system are acquired. According to the basic structure of the solutions, a complete solution classification corresponding to the initial data is given, including 121 categories. The results are verified by direct numerical simulation.Modulation instability gain and nonlinear modes generation in discrete cubic-quintic nonlinear Schrödinger equationhttps://zbmath.org/1516.353802023-09-19T14:22:37.575876Z"Abbagari, Souleymanou"https://zbmath.org/authors/?q=ai:abbagari.souleymanou"Houwe, Alphonse"https://zbmath.org/authors/?q=ai:houwe.alphonse"Saliou, Youssoufa"https://zbmath.org/authors/?q=ai:saliou.youssoufa"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanre"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi"Bouetou, Thomas Bouetou"https://zbmath.org/authors/?q=ai:bouetou-bouetou.thomasSummary: In this paper, we examined the effects of the nonlinear parameters and driven amplitude on the training envelope in discrete cubic-quintic nonlinear Schrödinger equation with arbitrarily high-order nonlinearities. From the numerical simulation, we exhibited the generation of the train of pulses and dark soliton when the driven amplitude is above the threshold. For a specific time of propagation, we illustrate how the variation of the driven amplitude and the nonlinear cubic-quintic parameters can generate instability in the forbidden gap. It emerges that the model of the discrete nonlinear Schrödinger equation with arbitrarily high-order nonlinearities can be used to generate the train of waves despite the fact that it is not integrable in the continuum limit approximation. We also displayed the effects of the cubic-quintic terms on the modulation instability growth rate. The obtained results will open new features to the train of pulses and dark soliton in the nonlinear fibers optics.Finite-element domain approximation for Maxwell variational problems on curved domainshttps://zbmath.org/1516.354012023-09-19T14:22:37.575876Z"Aylwin, Rubén"https://zbmath.org/authors/?q=ai:aylwin.ruben"Jerez-Hanckes, Carlos"https://zbmath.org/authors/?q=ai:jerez-hanckes.carlosSummary: We consider the problem of domain approximation in finite element methods for Maxwell equations on general curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest and an exact parametrization of the surface may not be readily available. In such cases, one is forced to approximate the domain by a sequence of polyhedral domains arising from inexact mesh. We deduce conditions on the quality of these approximations that ensure rates of error convergence between discrete solutions -- in the approximate domains -- to the continuous one in the original domain. Moreover, we present numerical results validating our claims.Analysis and simulation of an adhesive contact problem governed by fractional differential hemivariational inequalities with history-dependent operatorhttps://zbmath.org/1516.354052023-09-19T14:22:37.575876Z"Xuan, Hailing"https://zbmath.org/authors/?q=ai:xuan.hailing"Cheng, Xiaoliang"https://zbmath.org/authors/?q=ai:cheng.xiaoliangSummary: The main idea of this manuscript is to study an adhesive contact problem with long memory which is governed by a hemivariational inequality and a fractional differential equation. We first prove the existence of a unique solution of the fractional differential hemivariational inequality system. Subsequently, we consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. To the tail of this manuscript, we present two numerical simulation examples for the adhesive contact problem, which provide numerical evidence to support our theoretical predictions.On the equilibrium of the Poisson-Nernst-Planck-Bikermann model equipping with the steric and correlation effectshttps://zbmath.org/1516.354272023-09-19T14:22:37.575876Z"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo"Tang, Yijia"https://zbmath.org/authors/?q=ai:tang.yijia"Zhao, Yu"https://zbmath.org/authors/?q=ai:zhao.yuSummary: The Poisson-Nernst-Planck-Bikermann (PNPB) model, in which the ions and water molecules are treated as different species with non-uniform sizes and valences with interstitial voids, can describe the steric and correlation effects in ionic solution neglected by the Poisson-Nernst-Planck and Poisson-Boltzmann theories with point charge assumption. In the PNPB model, the electric potential is governed by the fourth-order Poisson-Bikermann (4PBik) equation instead of the Poisson equation so that it can describe the correlation effect. Moreover, the steric potential is included in the ionic and water fluxes as well as the equilibrium Fermi-like distributions which characterizes the steric effect quantitatively. In this work, we analyze the self-adjointness and the kernel of the fourth-order operator of the 4PBik equation. Also, we show the positivity of the void volume function and the convexity of the free energy. Following these properties, the well-posedness of the PNPB model in equilibrium is given. Furthermore, because the PNPB model has an energy dissipated structure, we adopt a finite volume scheme which preserves the energy dissipated property at the semi-discrete level. Various numerical investigations are given to show the parameter dependence of the steric effect to the steady state.Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditionershttps://zbmath.org/1516.354322023-09-19T14:22:37.575876Z"Pinzon, Jose"https://zbmath.org/authors/?q=ai:pinzon.jose"Siebenborn, Martin"https://zbmath.org/authors/?q=ai:siebenborn.martinSummary: In this article we propose a scalable shape optimization algorithm which is tailored for large scale problems and geometries represented by hierarchically refined meshes. Weak scalability and grid independent convergence is achieved via a combination of multigrid schemes for the simulation of the PDEs and quasi Newton methods on the optimization side. For this purpose a self-adapting, nonlinear extension operator is proposed within the framework of the method of mappings. This operator is demonstrated to identify critical regions in the reference configuration where geometric singularities have to arise or vanish. Thereby the set of admissible transformations is adapted to the underlying shape optimization situation. The performance of the proposed method is demonstrated for the example of drag minimization of an obstacle within a stationary, incompressible Navier-Stokes flow.Analytical solution of generalized diffusion-like equation of fractional orderhttps://zbmath.org/1516.354762023-09-19T14:22:37.575876Z"Shokhanda, Rachana"https://zbmath.org/authors/?q=ai:shokhanda.rachana"Goswami, Pranay"https://zbmath.org/authors/?q=ai:goswami.pranay"Nápoles Valdés, Juan E."https://zbmath.org/authors/?q=ai:napoles-valdes.juan-eSummary: In this paper, we consider a non-linear fractional diffusionlike equation. The existence and uniqueness of the solution of this type of equation are investigated. After that we use the homotopy perturbation method (HPM) to solve this equation by approximating a nonlinear function in a Taylor's series form and obtain an approximate solution.Fractional Tikhonov regularization method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equationhttps://zbmath.org/1516.354792023-09-19T14:22:37.575876Z"Wen, Jin"https://zbmath.org/authors/?q=ai:wen.jin"Yue, Chong-Wang"https://zbmath.org/authors/?q=ai:yue.chong-wang"Liu, Zhuan-Xia"https://zbmath.org/authors/?q=ai:liu.zhuan-xia"Wang, Shi-Juan"https://zbmath.org/authors/?q=ai:wang.shi-juanSummary: This paper is concerned with the problem of identifying the space-dependent source term and initial value simultaneously for a time-fractional diffusion equation. The inverse problem is ill-posed, and the idea of decoupling it into two operator equations is applied. In order to solve this inverse problem, a fractional Tikhonov regularization method is proposed. Furthermore, the corresponding convergence estimates are presented by using the a priori and a posteriori parameter choice rules. Several numerical examples compared with the classical Tikhonov regularization are constructed for verifying the accuracy and efficiency of the proposed method.Maximum rate of growth of enstrophy in solutions of the fractional Burgers equationhttps://zbmath.org/1516.354812023-09-19T14:22:37.575876Z"Yun, Dongfang"https://zbmath.org/authors/?q=ai:yun.dongfang"Protas, Bartosz"https://zbmath.org/authors/?q=ai:protas.bartoszSummary: This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blowup in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutionshttps://zbmath.org/1516.354862023-09-19T14:22:37.575876Z"Agnelli, J. P."https://zbmath.org/authors/?q=ai:agnelli.juan-pablo"De Cezaro, A."https://zbmath.org/authors/?q=ai:de-cezaro.adriano"Leitão, A."https://zbmath.org/authors/?q=ai:leitao.antonioSummary: We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values.
Following \textit{A. De Cezaro} et al. [Inverse Probl. 29, No. 1, Article ID 015003, 23 p. (2013; Zbl 1302.65147)], a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method.
We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al.'s work [loc. cit.], proving convergence and stability of the proposed parameter identification method.
A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in [\textit{J. P. Agnelli} et al., ESAIM, Control Optim. Calc. Var. 23, No. 2, 663--683 (2017; Zbl 1358.49031)] demonstrating the effectiveness of this primal-dual algorithm.The ``exterior approach'' applied to the inverse obstacle problem for the heat equationhttps://zbmath.org/1516.354992023-09-19T14:22:37.575876Z"Bourgeois, Laurent"https://zbmath.org/authors/?q=ai:bourgeois.laurent"Dardé, Jérémi"https://zbmath.org/authors/?q=ai:darde.jeremiSummary: In this paper we consider the ``exterior approach'' to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification.Regularization and stability estimates for an inverse source problem of the radially symmetric parabolic equationhttps://zbmath.org/1516.355022023-09-19T14:22:37.575876Z"Cheng, Wei"https://zbmath.org/authors/?q=ai:cheng.weiSummary: We consider an inverse problem of determining an unknown source term in the radially symmetric parabolic equation from a noisy final data and prove the uniqueness of solution for the problem. Using the Hölder inequality, we obtain a conditional stability for the space-dependent source term. A modified quasi-reversibility method is applied to deal with the ill-posedness of the problem. A Hölder-type error estimate between the approximate solution and the exact solution is provided by introducing some technical inequalities and choosing a suitable regularization parameter.Iterative methods for photoacoustic tomography in attenuating acoustic mediahttps://zbmath.org/1516.355132023-09-19T14:22:37.575876Z"Haltmeier, Markus"https://zbmath.org/authors/?q=ai:haltmeier.markus"Kowar, Richard"https://zbmath.org/authors/?q=ai:kowar.richard"Nguyen, Linh V."https://zbmath.org/authors/?q=ai:nguyen.linh-viet|nguyen-viet-linh.Summary: The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as an ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. In contrast to time reversal, the employed adjoint wave equation is again damping and, thus has a stable solution. This stability property can be clearly seen in our numerical results. Moreover, the presented numerical results clearly demonstrate the efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case.An \textit{a posteriori} Fourier regularization method for identifying the unknown source of the space-fractional diffusion equationhttps://zbmath.org/1516.355262023-09-19T14:22:37.575876Z"Li, Xiao-Xiao"https://zbmath.org/authors/?q=ai:li.xiaoxiao"Lei, Jin Li"https://zbmath.org/authors/?q=ai:lei.jin-li"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1Summary: In this paper, we identify the unknown source which depends only on spatial variable for a fractional diffusion equation using the Fourier method. Not alike the previous literature, we propose to choose the regularization parameter by an \textit{a posteriori} rule, with which we can obtain error estimate of Hölder type between the exact solution and the regularized approximation. Numerical simulations show that the proposed scheme is effective and stable.Semianalytical method for the identification of inclusions by air-cored coil interaction in ferromagnetic mediahttps://zbmath.org/1516.355412023-09-19T14:22:37.575876Z"Vafeas, Panayiotis"https://zbmath.org/authors/?q=ai:vafeas.panayiotis"Skarlatos, Anastassios"https://zbmath.org/authors/?q=ai:skarlatos.anastassios"Theodoulidis, Theodoros"https://zbmath.org/authors/?q=ai:theodoulidis.theodoros-p"Lesselier, Dominique"https://zbmath.org/authors/?q=ai:lesselier.dominiqueSummary: The magnetostatic harmonic fields scattered by a near-surface air inclusion of arbitrary shape, embedded in a conductive ferromagnetic medium and illuminated by a current-carrying coil, are investigated. The scattering domain is separated into homogeneous subdomains under the assumption of a suitable truncation at a long distance from the incident source, whereas a perfect magnetic boundary condition is implied. The introduced methodology addresses the full coupling between the two interfaces, ie, the plane that distinguishes the half-space ferromagnetic material from the open air and the arbitrary surface among the inclusion and the ferromagnetic region. Therein, continuity conditions are applied in a rigorous way, while the expected behavior of the fields, either as ascending or as descending, are taken into account. The potentials associated with the half-space are expanded via cylindrical harmonic eigenfunctions, while those related with the inclusion's arbitrary geometry admit generalized-type formalism. However, since the transmission conditions involve potentials with different eigenexpansions, we are obliged to rewrite cylindrical to generalized functions and vice versa, obtaining handy relationships in terms of easy-to-handle integrals, where orthogonality then is feasible. Once done, the calculation of the exact solutions leads to infinite linear algebraic systems, manipulated through standard cut-off techniques. Thus, we obtain the implicated fields in a general analytical and compact fashion, independent of the inclusion's geometry. We demonstrate the efficiency of the analytical model approach, assuming the degenerate case of a spherical inclusion, whereas the air-cored coil simulation via a numerical procedure validates our method. The calculation is very fast, rendering it suitable for use with parametric inversion algorithms.The simplified Tikhonov regularization method for solving a Riesz-Feller space-fractional backward diffusion problemhttps://zbmath.org/1516.355442023-09-19T14:22:37.575876Z"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1"Li, Xiao-Xiao"https://zbmath.org/authors/?q=ai:li.xiaoxiao"Li, Dun-Gang"https://zbmath.org/authors/?q=ai:li.dungang"Wang, Lan"https://zbmath.org/authors/?q=ai:wang.lan.1Summary: In this paper, we consider a backward diffusion problem for a space-fractional diffusion equation. Such a problem is obtained from the classical diffusion equation in which the second-order space derivative is replaced with a Riesz-Feller derivative of order \(\alpha \in (0,2]\). The temperature is sought from a measured temperature history at a fixed time \(t=T\). This problem is ill-posed, i.e., the solution (if it exists) does not depend on the data. The simplified Tikhonov regularization method is proposed to solve this problem. Under the \textit{a priori} bound assumptions for the exact solution, the convergence estimate is presented. Moreover, a \textit{posteriori} parameter choice rule is proposed and the convergence estimate is also obtained. All estimates are Hölder type. Numerical examples are presented to illustrate the validity and effectiveness of this method.Bayesian inversion of log-normal eikonal equationshttps://zbmath.org/1516.355452023-09-19T14:22:37.575876Z"Yeo, Zhan Fei"https://zbmath.org/authors/?q=ai:yeo.zhan-fei"Hoang, Viet Ha"https://zbmath.org/authors/?q=ai:hoang.viet-haSummary: We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen-Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstructionhttps://zbmath.org/1516.355462023-09-19T14:22:37.575876Z"Yu, Jie"https://zbmath.org/authors/?q=ai:yu.jie"Liu, Yikan"https://zbmath.org/authors/?q=ai:liu.yikan"Yamamoto, Masahiro"https://zbmath.org/authors/?q=ai:yamamoto.masahiroSummary: In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with either partial boundary or interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.Timing of transients: quantifying reaching times and transient behavior in complex systemshttps://zbmath.org/1516.371362023-09-19T14:22:37.575876Z"Kittel, Tim"https://zbmath.org/authors/?q=ai:kittel.tim"Heitzig, Jobst"https://zbmath.org/authors/?q=ai:heitzig.jobst"Webster, Kevin"https://zbmath.org/authors/?q=ai:webster.kevin-n"Kurths, Jürgen"https://zbmath.org/authors/?q=ai:kurths.jurgenSummary: In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, area under distance curve and regularized reaching time, that capture two complementary aspects of transient dynamics. The first, area under distance curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are `reluctant', i.e. stay distant from the attractor for long, or `eager' to approach it right away. Regularized reaching time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much `earlier' or `later' than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.Improving series convergence: the simple pendulum and beyondhttps://zbmath.org/1516.400022023-09-19T14:22:37.575876Z"Duki, Solomon F."https://zbmath.org/authors/?q=ai:duki.solomon-fekade"Doerr, T. P."https://zbmath.org/authors/?q=ai:doerr.t-p"Yu, Yi-Kuo"https://zbmath.org/authors/?q=ai:yu.yikuoSummary: A simple and easy to implement method for improving the convergence of a power series is presented. We observe that the most obvious or analytically convenient point about which to make a series expansion is not always the most computationally efficient. Series convergence can be dramatically improved by choosing the center of the series expansion to be at or near the average value at which the series is to be evaluated. For illustration, we apply this method to the well-known simple pendulum and to the Mexican hat type of potential. Large performance gains are demonstrated. While the method is not always the most computationally efficient on its own, it is effective, straightforward, quite general, and can be used in combination with other methods.Comment on: ``Improving series convergence: the simple pendulum and beyond''https://zbmath.org/1516.400032023-09-19T14:22:37.575876Z"Fernández, Francisco M."https://zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: In this comment we analyze the improved series proposed by \textit{S. F. Duki} et al. [Eur. J. Phys. 39, No. 6, Article ID 065802, 8 p. (2018; Zbl 1516.40002)] for the approximate calculation of a variety of physical quantities like the period of the simple pendulum and an integral of the exponential of a two-dimensional potential-energy function. We show that the application of the approach to the latter case is unnecessary because both the expansion coefficients and the original problem are given in terms of a similar error function. Present results for the straightforward expansion with exact analytic coefficients do not show the oscillatory behaviour exhibited by the calculations of those authors. We give reasons why the improved approach of Duki et al. [loc. cit.] may not be suitable for the treatment of many realistic physical problems.Implicit surface reconstruction with a curl-free radial basis function partition of unity methodhttps://zbmath.org/1516.410012023-09-19T14:22:37.575876Z"Drake, Kathryn P."https://zbmath.org/authors/?q=ai:drake.kathryn-p"Fuselier, Edward J."https://zbmath.org/authors/?q=ai:fuselier.edward-j"Wright, Grady B."https://zbmath.org/authors/?q=ai:wright.grady-bApproximations coming from smooth surfaces with implicit definitions (being often in three dimensions) are a very successful tool in computer aided geometric design. They are the subject-matter of this article.
At the same time, there are many highly accurate forms of approximation (using interpolation or quasi-interpolation) with the so-called radial basis functions that are used by the authors, and they provide an interesting combination with the mentioned implicit surface representations. A certain curl-free variant of these radial basis function is particularly suitable for implicitly defined surfaces and approximations employing them. This is relevant in this paper, because the normals of the sought surface are curl-free and the mentioned surface will be defined by just those normals. Of course, the algorithms are going to be particularly attractive if scattered data are admitted, and indeed those are allowed in all approximations methods with radial basis functions and also here explicitly in this paper.
The approximations of spheres are combined in the article with a version of the well-known partition of unity (PoU) method to make the results numerically stable and efficient. Many three-dimensional examples are provided by the authors.
Reviewer: Martin D. Buhmann (Gießen)Compact cubic splineshttps://zbmath.org/1516.410062023-09-19T14:22:37.575876Z"Corless, Robert M."https://zbmath.org/authors/?q=ai:corless.robert-m"Sevyeri, Leili Rafiee"https://zbmath.org/authors/?q=ai:sevyeri.leili-rafieeA new way to form cubic splines that provide interpolants with exactness of degree four is introduced. These cubic spline interpolants are calculated by using tridiagonal matrices which are totally nonnegative and positive definite, and those are very attractive features for the numerical methods.
The new cubic splines have the advantage that fourth order accuracy is guaranteed without extra work at the two end-points of the interval, so no free knots, or not-a-knot conditions, are needed. Therefore the end-points are treated much more efficiently. Their coefficients come from so-called compact divided differences, and those are computed by solving a certain complex integral.
Reviewer: Martin D. Buhmann (Gießen)Estimates of solutions to infinite systems of linear equations and the problem of interpolation by cubic splines on the real linehttps://zbmath.org/1516.410082023-09-19T14:22:37.575876Z"Volkov, Yu. S."https://zbmath.org/authors/?q=ai:volkov.yuri-s"Novikov, S. I."https://zbmath.org/authors/?q=ai:novikov.sergey-iDiagonally dominant matrices often turn up when spline spaces are used to produce interpolants to functions and other approximants. Useful examples for such applications come from approximations using cubic splines.
In this paper, bi-infinite matrices (e.g., Toeplitz matrices) are studied which are diagonally dominant, and estimates on norms are provided. In particular, the existence of splines of degree three is shown for infinitely many knots where there are no constraints on the knot distances. In this fashion, not only the existence and uniqueness of cubic spline interpolants for infinitely many knots of arbitrary spacing is shown, but the familiar error estimates are carried over from the finite to the bi-infinite case of spline interpolation degree 3.
Reviewer: Martin D. Buhmann (Gießen)Generalized matrix spectral factorization with symmetry and applications to symmetric quasi-tight frameletshttps://zbmath.org/1516.420262023-09-19T14:22:37.575876Z"Diao, Chenzhe"https://zbmath.org/authors/?q=ai:diao.chenzhe"Han, Bin"https://zbmath.org/authors/?q=ai:han.bin|han.bin.1"Lu, Ran"https://zbmath.org/authors/?q=ai:lu.ranMotivated by the recent development of quasi-tight framelets [\textit{C. Diao} and \textit{B. Han}, Appl. Comput. Harmon. Anal. 49, No. 1, 123--151 (2020; Zbl 1437.42052)], the authors study and characterize generalized spectral factorizations with symmetry for \(2 \times 2\) matrices of Laurent polynomials. They then use these results to find a necessary and sufficient condition for the existence of symmetric quasi--tight framelets with two generators. The proofs of all the main results are constructive and can therefore be used as algorithms. The authors provide several examples to illustrate their theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry. The paper is organized as follows: Section 1 is titled ``Introduction, motivations and main results''. Section 2 briefly reviews some necessary background on framelets, explains the reasons for considering quasi-tight framelets with symmetry and provides some illustrative examples. Section 3 considers two special cases of their main theorem (Theorem 1.1), on the factorization of a \(2 \times 2\) matrix of Laurent polynomials with symmetry, before proving it in full generality. Finally, in Section 4, Theorem 1.1 is used to prove Theorem 1.3, which characterizes quasi-tight framelet filter banks with symmetry, and their associated symmetric quasi-tight framelets in \(L_2(R)\).
Reviewer: Richard A. Zalik (Auburn)Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaceshttps://zbmath.org/1516.471082023-09-19T14:22:37.575876Z"Hu, Shaotao"https://zbmath.org/authors/?q=ai:hu.shaotao"Wang, Yuanheng"https://zbmath.org/authors/?q=ai:wang.yuanheng.1|wang.yuanheng"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing.1"Wang, Fenghui"https://zbmath.org/authors/?q=ai:wang.fenghuiSummary: In this paper, we propose a new inertial viscosity iterative algorithm for solving the variational inequality problem with a pseudo-monotone operator and the fixed point problem involving a nonexpansive mapping in real Hilbert spaces. The advantage of the proposed algorithm is that it can work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence of the sequence generated by the proposed algorithm is proved under some suitable assumptions imposed on the parameters. Some numerical experiments are given to support our main results.Operator splitting for dissipative delay equationshttps://zbmath.org/1516.471192023-09-19T14:22:37.575876Z"Bátkai, András"https://zbmath.org/authors/?q=ai:batkai.andras"Csomós, Petra"https://zbmath.org/authors/?q=ai:csomos.petra"Farkas, Bálint"https://zbmath.org/authors/?q=ai:farkas.balintSummary: We investigate Lie-Trotter product formulae for abstract nonlinear evolution equations with delay. Using results from the theory of nonlinear contraction semigroups in Hilbert spaces, we explain the convergence of the splitting procedure. The order of convergence is also investigated in detail, and some numerical illustrations are presented.Stability of the solution set of quasi-variational inequalities and optimal controlhttps://zbmath.org/1516.490042023-09-19T14:22:37.575876Z"Alphonse, Amal"https://zbmath.org/authors/?q=ai:alphonse.amal"Hintermüller, Michael"https://zbmath.org/authors/?q=ai:hintermuller.michael"Rautenberg, Carlos N."https://zbmath.org/authors/?q=ai:rautenberg.carlos-nSummary: For a class of quasi-variational inequalities (QVIs) of obstacle type, the stability of its solution set and associated optimal control problems are considered. These optimal control problems are nonstandard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to monotone perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.Optimal control of time-fractional convection-diffusion-reaction problem employing compact integrated RBF methodhttps://zbmath.org/1516.490122023-09-19T14:22:37.575876Z"Mahmoudi, Mahmoud"https://zbmath.org/authors/?q=ai:mahmoudi.mahmoud"Shojaeizadeh, Tahereh"https://zbmath.org/authors/?q=ai:shojaeizadeh.tahereh"Darehmiraki, Majid"https://zbmath.org/authors/?q=ai:darehmiraki.majidSummary: Optimal control has always been investigated for various systems in order to optimize them. It has also been considered as well within convection-diffusion-reaction equations utilizing different methods. The present study introduces a new method for solving the optimal control problem of time-fractional convection-diffusion-reaction using local compact integrated radial basis function (CIRBF). The main idea is to discretize the spatial derivatives by compounding the integrated RBF and compact approach in a stencil. At first, the time-fractional part of the problem is discretized by using the shifted Grünwald formulas. Then, the first- and second-order derivatives of the state and adjoint variables at the nodal points are approximated by the CIRBF method. Moreover, the stability and convergence of the procedure have been proved. Finally, some examples are presented to illustrate the efficiency and accuracy of the process.Error estimates of Fourier finite volume element method for parabolic Dirichlet boundary optimal control problems on complex connected domainshttps://zbmath.org/1516.490282023-09-19T14:22:37.575876Z"Liu, Wenju"https://zbmath.org/authors/?q=ai:liu.wenju"Zhao, Tengjin"https://zbmath.org/authors/?q=ai:zhao.tengjin"Ito, Kazufumi"https://zbmath.org/authors/?q=ai:ito.kazufumi"Zhang, Zhiyue"https://zbmath.org/authors/?q=ai:zhang.zhiyueSummary: In this paper, we study the parabolic Dirichlet boundary optimal control on complex connected domains. It is well known that both the complex connected domain and the Dirichlet boundary control bring great difficulties to theoretical analysis and numerical calculation. The complex connected domain is a typical non-convex domain, and it is difficult for the traditional numerical method to obtain the same convergence order as the state and adjoint state for the Dirichlet boundary control. The optimal system of the proposed control problem is first obtained by using Lagrange multiplier method. Then, based on the variational form, the Fourier finite volume element method is used to obtain the fully discrete scheme of the optimality system, that is, using Fourier expansion method in azimuthal direction and applying finite volume element method in radial direction respectively, and the Crank-Nicolson scheme in the time direction. Next, we strictly prove the error estimates of state, adjoint state and Dirichlet boundary control by using the variational discretization concept. Finally, the theoretical analysis results and the feasibility and effectiveness of the proposed method are verified by numerical experiments.A global optimization approach for multimarginal optimal transport problems with Coulomb costhttps://zbmath.org/1516.490292023-09-19T14:22:37.575876Z"Hu, Yukuan"https://zbmath.org/authors/?q=ai:hu.yukuan"Chen, Huajie"https://zbmath.org/authors/?q=ai:chen.huajie"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.6|liu.xin.8|liu.xin.5|liu.xin.13|liu.xin.4|liu.xin.9|liu.xin.1|liu.xin.10|liu.xin.7Summary: In this work, we construct a novel numerical method for solving the multimarginal optimal transport problems with Coulomb cost. This type of optimal transport problem arises in quantum physics and plays an important role in understanding the strongly correlated quantum systems. With a Monge-like ansatz, we transfer the original high-dimensional problems into mathematical programmings with generalized complementarity constraints, and thus the curse of dimensionality is surmounted. However, the latter ones are themselves hard to deal with from both theoretical and practical perspectives. Moreover, in the presence of nonconvexity, brute-force searching for global solutions becomes prohibitive as the problem size grows large. To this end, we propose a global optimization approach for solving the nonconvex optimization problems, by exploiting an efficient proximal block coordinate descent local solver and an initialization subroutine based on hierarchical grid refinements. We conduct numerical simulations on some typical physical systems to show the efficiency of our approach. The results match well with both theoretical predictions and physical intuitions and provide indications for Monge solutions in two-dimensional contexts. In addition, we give the first visualization of approximate optimal transport maps for some two-dimensional systems.Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equationhttps://zbmath.org/1516.490302023-09-19T14:22:37.575876Z"Kerkhoff, Xenia"https://zbmath.org/authors/?q=ai:kerkhoff.xenia"May, Sandra"https://zbmath.org/authors/?q=ai:may.sandraSummary: We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.Continuity with respect to the speed for optimal ship forms based on Michell's formulahttps://zbmath.org/1516.490402023-09-19T14:22:37.575876Z"Dambrine, Julien"https://zbmath.org/authors/?q=ai:dambrine.julien"Pierre, Morgan"https://zbmath.org/authors/?q=ai:pierre.morganSummary: We consider a ship hull design problem based on Michell's wave resistance. The half hull is represented by a nonnegative function and we seek the function whose domain of definition has a given area and which minimizes the total resistance for a given speed and a given volume. We show that the optimal hull depends only on two parameters without dimension, the viscous drag coefficient and the Froude number of the area of the support. We prove that, up to uniqueness, the optimal hull depends continuously on these two parameters. Moreover, the contribution of Michell's wave resistance vanishes as either the Froude number or the drag coefficient goes to infinity. Numerical simulations confirm the theoretical results for large Froude numbers. For Froude numbers typically smaller than 1, the famous bulbous bow is numerically recovered. For intermediate Froude numbers, a ``sinking'' phenomenon occurs. It can be related to the nonexistence of a minimizer.Inexact infinite products of weak quasi-contraction mappings in \(b\)-metric spaceshttps://zbmath.org/1516.540282023-09-19T14:22:37.575876Z"Gupta, Anuradha"https://zbmath.org/authors/?q=ai:gupta.anuradha"Rohilla, Manu"https://zbmath.org/authors/?q=ai:rohilla.manuThe influence of errors on the convergence of infinite products of weak quasi-contraction mappings in $b$-metric spaces is explored. An example demonstrating the necessity of convergence of the sequence of computational errors to zero is also provided. Also, the authors discuss weak ergodic theorems in the setting of $b$-metric spaces.
Reviewer: Zoran D. Mitrović (Banja Luka)Existence and approximation of fixed points of \(\lambda\)-hybrid mappings in complete CAT(0) spaceshttps://zbmath.org/1516.540332023-09-19T14:22:37.575876Z"Khoshhal, Mahmood"https://zbmath.org/authors/?q=ai:khoshhal.mahmood"Bagha, Davood Ebrahimi"https://zbmath.org/authors/?q=ai:ebrahimi-bagha.davoodThe authors introduce a class of \(\lambda\)-hybrid mappings to prove some general existence fixed point theorems in Hadamard spaces. Also, a convergence result is established for \(\lambda\)-hybrid mappings using Mann's iterative procedure.
Reviewer: Memudu Olaposi Olatinwo (Ile-Ife)A fixed point theorem for a pair of generalized nonexpansive mappings in uniformly convex metric spaceshttps://zbmath.org/1516.540602023-09-19T14:22:37.575876Z"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao.7"Fan, Honglei"https://zbmath.org/authors/?q=ai:fan.hongleiSummary: In uniformly convex metric spaces, we study the existence and uniqueness of a common fixed point for a pair of generalized nonexpansive mappings with some weak conditions. Meanwhile, we introduce a new Krasnoselskii type iterative algorithm for approximating the common fixed point. A numerical example is also given to demonstrate the main result. Our results generalize and improve some recent corresponding results.On mixed fractional stochastic differential equations with discontinuous drift coefficienthttps://zbmath.org/1516.600342023-09-19T14:22:37.575876Z"Sönmez, Ercan"https://zbmath.org/authors/?q=ai:sonmez.ercanSummary: We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.State-dependent importance sampling for estimating expectations of functionals of sums of independent random variableshttps://zbmath.org/1516.620062023-09-19T14:22:37.575876Z"Ben Amar, Eya"https://zbmath.org/authors/?q=ai:ben-amar.eya"Ben Rached, Nadhir"https://zbmath.org/authors/?q=ai:ben-rached.nadhir"Haji-Ali, Abdul-Lateef"https://zbmath.org/authors/?q=ai:haji-ali.abdul-lateef"Tempone, Raúl"https://zbmath.org/authors/?q=ai:tempone.raul-fSummary: Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.Learning-based importance sampling via stochastic optimal control for stochastic reaction networkshttps://zbmath.org/1516.620072023-09-19T14:22:37.575876Z"Ben Hammouda, Chiheb"https://zbmath.org/authors/?q=ai:ben-hammouda.chiheb"Ben Rached, Nadhir"https://zbmath.org/authors/?q=ai:ben-rached.nadhir"Tempone, Raúl"https://zbmath.org/authors/?q=ai:tempone.raul-f"Wiechert, Sophia"https://zbmath.org/authors/?q=ai:wiechert.sophiaSummary: We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.Fitting Matérn smoothness parameters using automatic differentiationhttps://zbmath.org/1516.620142023-09-19T14:22:37.575876Z"Geoga, Christopher J."https://zbmath.org/authors/?q=ai:geoga.christopher-j"Marin, Oana"https://zbmath.org/authors/?q=ai:marin.oana"Schanen, Michel"https://zbmath.org/authors/?q=ai:schanen.michel"Stein, Michael L."https://zbmath.org/authors/?q=ai:stein.michael-lSummary: The Matérn covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter \(\nu\) gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Matérn covariance function with respect to \(\nu\) require derivatives of the modified second-kind Bessel function \(\mathcal{K}_\nu\) with respect to \(\nu\). While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing \(\nu\) as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating \(\nu\) use finite difference estimates for \(\partial_\nu\mathcal{K}_\nu\). In this work, we introduce a new implementation of \(\mathcal{K}_\nu\) that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. We provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.On the optimality of the Oja's algorithm for online PCAhttps://zbmath.org/1516.620182023-09-19T14:22:37.575876Z"Liang, Xin"https://zbmath.org/authors/?q=ai:liang.xinSummary: In this paper we analyze the behavior of the Oja's algorithm for online/streaming principal component subspace estimation. It is proved that with high probability it performs an efficient, gap-free, global convergence rate to approximate an principal component subspace for any sub-Gaussian distribution. Moreover, it is the first time to show that the convergence rate, namely the upper bound of the approximation, exactly matches the lower bound of an approximation obtained by the offline/classical PCA up to a constant factor.Multi-index antithetic stochastic gradient algorithmhttps://zbmath.org/1516.620192023-09-19T14:22:37.575876Z"Majka, Mateusz B."https://zbmath.org/authors/?q=ai:majka.mateusz-b"Sabate-Vidales, Marc"https://zbmath.org/authors/?q=ai:sabate-vidales.marc"Szpruch, Łukasz"https://zbmath.org/authors/?q=ai:szpruch.lukaszSummary: Stochastic Gradient Algorithms (SGAs) are ubiquitous in computational statistics, machine learning and optimisation. Recent years have brought an influx of interest in SGAs, and the non-asymptotic analysis of their bias is by now well-developed. However, relatively little is known about the optimal choice of the random approximation (e.g mini-batching) of the gradient in SGAs as this relies on the analysis of the variance and is problem specific. While there have been numerous attempts to reduce the variance of SGAs, these typically exploit a particular structure of the sampled distribution by requiring a priori knowledge of its density's mode. In this paper, we construct a Multi-index Antithetic Stochastic Gradient Algorithm (MASGA) whose implementation is independent of the structure of the target measure. Our rigorous theoretical analysis demonstrates that for log-concave targets, MASGA achieves performance on par with Monte Carlo estimators that have access to unbiased samples from the distribution of interest. In other words, MASGA is an optimal estimator from the mean square error-computational cost perspective within the class of Monte Carlo estimators. To illustrate the robustness of our approach, we implement MASGA also in some simple non-log-concave numerical examples, however, without providing theoretical guarantees on our algorithm's performance in such settings.Circular designs for total effects under interference modelshttps://zbmath.org/1516.620722023-09-19T14:22:37.575876Z"Kong, Xiangshun"https://zbmath.org/authors/?q=ai:kong.xiangshun"Zhang, Xueru"https://zbmath.org/authors/?q=ai:zhang.xueru"Zheng, Wei"https://zbmath.org/authors/?q=ai:zheng.weiSummary: This paper studies circular designs for interference models, where a treatment assigned to a plot also affects its neighboring plots within a block. For the purpose of estimating total effects, the circular neighbor balanced design was shown to be universally optimal among designs which do not allow treatments to be neighbors of themselves. Our study shows that self-neighboring block sequences are actually the main ingredient for an optimal design. Here, we adopt the approximate design framework and study optimal designs in the whole design space. Our approach is flexible enough to accommodate all possible design parameters, that is the block size and the number of blocks and treatments. This approach can be broken down into two main steps: the identification of the minimal supporting set of block sequences and the optimality condition built on it. The former is critical for reducing the computational time from almost infinity to seconds. Meanwhile, the task of finding the minimal set is normally achieved through numerical methods, which can only handle small block sizes. Our approach is of a hybrid nature in order to deal with all design sizes. When block size is not large, we provide explicit expressions of the minimal set instead of relying on numerical methods. For larger block sizes when a typical numerical method would fail, we theoretically derived a reasonable size intermediate set of sequences, from which the minimal set can be quickly derived through a customized algorithm. Taking it further, the optimality conditions allow us to obtain both symmetric and asymmetric designs. Lastly, we also investigate the trade-off issue between circular and noncircular designs, and provide guidelines on the choices.A numerical scheme for solving a class of time fractional integro-partial differential equations with Caputo-Fabrizio derivativehttps://zbmath.org/1516.620962023-09-19T14:22:37.575876Z"Mohammadpour, A."https://zbmath.org/authors/?q=ai:mohammadpour.alireza|mohammadpour.adel"Babaei, A."https://zbmath.org/authors/?q=ai:babaei.ardavan|babaei.amin|babaei.alireza|babaei.abbas|babaei.afshin"Banihashemi, S."https://zbmath.org/authors/?q=ai:banihashemi.shokoofeh|banihashemi.seddigheh(no abstract)Third multigrid seminar. Papers from the seminar, Biesenthal, Germany, May 2--6, 1988https://zbmath.org/1516.650012023-09-19T14:22:37.575876ZThe articles of this volume will be reviewed individually within the serial ``Rep., Akad. Wiss. DDR, Karl-Weierstrass-Inst. Math. 03/89''.Nonmonotone globalization for Anderson acceleration via adaptive regularizationhttps://zbmath.org/1516.650022023-09-19T14:22:37.575876Z"Ouyang, Wenqing"https://zbmath.org/authors/?q=ai:ouyang.wenqing"Tao, Jiong"https://zbmath.org/authors/?q=ai:tao.jiong"Milzarek, Andre"https://zbmath.org/authors/?q=ai:milzarek.andre"Deng, Bailin"https://zbmath.org/authors/?q=ai:deng.bailinSummary: Anderson acceleration (\textsf{AA}) is a popular method for accelerating fixed-point iterations, but may suffer from instability and stagnation. We propose a globalization method for \textsf{AA} to improve stability and achieve unified global and local convergence. Unlike existing \textsf{AA} globalization approaches that rely on safeguarding operations and might hinder fast local convergence, we adopt a nonmonotone trust-region framework and introduce an adaptive quadratic regularization together with a tailored acceptance mechanism. We prove global convergence and show that our algorithm attains the same local convergence as \textsf{AA} under appropriate assumptions. The effectiveness of our method is demonstrated in several numerical experiments.Strong stability preserving multistep schemes for forward backward stochastic differential equationshttps://zbmath.org/1516.650032023-09-19T14:22:37.575876Z"Fang, Shuixin"https://zbmath.org/authors/?q=ai:fang.shuixin"Zhao, Weidong"https://zbmath.org/authors/?q=ai:zhao.weidong"Zhou, Tao"https://zbmath.org/authors/?q=ai:zhou.taoSummary: In this work, we are concerned with strong stability preserving multistep (SSPM) schemes for forward backward stochastic differential equations (FBSDEs). To this aim, we first perform a comprehensive analysis on a general type of multistep schemes for FBSDEs, based on which we present new sufficient conditions on the coefficients such that the associated schemes are stable and enjoy certain order of consistency. Upon these results, we propose a practical way to design high-order SSPM schemes for FBSDEs. Numerical experiments are carried out to demonstrate the strong stability of our SSPM schemes.A fast Euler-Maruyama method for Riemann-Liouville stochastic fractional nonlinear differential equationshttps://zbmath.org/1516.650042023-09-19T14:22:37.575876Z"Zhang, Jingna"https://zbmath.org/authors/?q=ai:zhang.jingna"Lv, Jingyun"https://zbmath.org/authors/?q=ai:lv.jingyun"Huang, Jianfei"https://zbmath.org/authors/?q=ai:huang.jianfei"Tang, Yifa"https://zbmath.org/authors/?q=ai:tang.yifaSummary: In this paper, based on the sum-of-exponentials approximation, a fast Euler-Maruyama (EM) method is constructed to solve a kind of multi-term Riemann-Liouville stochastic fractional differential equations. Then the strong convergence order \(\min\{1 - \alpha_m, 0.5\}\) of the proposed EM method is proved with Riemann-Liouville fractional derivatives' orders satisfying \(0 < \alpha_1 < \alpha_2 < \cdots < \alpha_m < 1\). Finally, two numerical examples are given to support the theoretical results and show the powerful computational performance of the fast EM method.Taylor type and Hermite type interpolants in \(\mathbb{R}^n \)https://zbmath.org/1516.650052023-09-19T14:22:37.575876Z"Phung Van Manh"https://zbmath.org/authors/?q=ai:phung-van-manh."Nguyen Van Trao"https://zbmath.org/authors/?q=ai:nguyen-van-trao."Phan Thanh Tung"https://zbmath.org/authors/?q=ai:phan-thanh-tung."Le Ngoc Cuong"https://zbmath.org/authors/?q=ai:le-ngoc-cuong.Summary: We construct new polynomial interpolation schemes of Taylor and Hermite types in \(\mathbb{R}^n \). The interpolation conditions are real parts and imaginary parts of certain differential operators. We also give formulas for the interpolation polynomials which are of Newton form and can be computed by an algorithm.Adapting cubic Hermite splines to the presence of singularities through correction termshttps://zbmath.org/1516.650062023-09-19T14:22:37.575876Z"Amat, Sergio"https://zbmath.org/authors/?q=ai:amat.sergio-p"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilin.1"Ruiz-Álvarez, Juan"https://zbmath.org/authors/?q=ai:ruiz-alvarez.juan"Solano, Concepción"https://zbmath.org/authors/?q=ai:solano.concepcion"Trillo, Juan C."https://zbmath.org/authors/?q=ai:trillo.juan-carlosSummary: Hermite interpolation is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This paper is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with \(O(h^4)\) accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained.Local linear independence of bilinear (and higher degree) B-splines on hierarchical T-mesheshttps://zbmath.org/1516.650072023-09-19T14:22:37.575876Z"Groiss, Lisa"https://zbmath.org/authors/?q=ai:groiss.lisa"Jüttler, Bert"https://zbmath.org/authors/?q=ai:juttler.bert"Pan, Maodong"https://zbmath.org/authors/?q=ai:pan.maodongSummary: We generate hierarchical T-meshes by repeatedly inserting new line segments, in order to adapt both the size and the shape of the cells to the specific requirements of the underlying application. The associated spaces of bilinear spline functions are spanned by the locally refined (LR) B-splines of \textit{T. Dokken} et al. [Comput. Aided Geom. Des. 30, No. 3, 331--356 (2013; Zbl 1264.41011)], which are products of univariate B-splines defined over local knot vectors. Our method guarantees that these functions are linearly independent for the obtained class of hierarchical T-meshes. Furthermore, they even possess the property of local linear independence, since exactly 4 functions take non-zero values on each cell of the mesh, and they form a partition of unity without the need to perform additional scaling or truncation.
The correctness of our refinement algorithm is verified by enumerating the (newly introduced) local configurations, which represent the possible topologies of the mesh in the vicinity of a cell. It is then shown that the method works well in all the resulting situations. The fairly large number (385) of cases sheds some light on the challenges associated with linear independence in the context of LR B-spline refinement.
Additionally we apply our method to meshes that possess the additional property local semi-regularity (first studied by \textit{F. Weller} and \textit{H. Hagen} [in: Mathematical methods for curves and surfaces. Papers from an international conference held June 16-21, 1994 in Ulvik, Norway. Nashville, TN: Vanderbilt University Press. 563--572 (1995; Zbl 0835.65011)]), where only 49 topologically different situations may arise. However, we observe that the resulting class of hierarchical T-meshes is less flexible, which leads to a substantial amount of excess refinement in applications.
Finally we note that the same construction can be applied to generate basis functions for \(C^s\)-smooth spline spaces of degree \(p = 2 s + 1\), again with the property of local linear independence.Sufficient condition for injectivity of NURBS volumes by tangent coneshttps://zbmath.org/1516.650082023-09-19T14:22:37.575876Z"Yu, Ying-Ying"https://zbmath.org/authors/?q=ai:yu.yingying"Ji, Ye"https://zbmath.org/authors/?q=ai:ji.ye"Zhu, Chun-Gang"https://zbmath.org/authors/?q=ai:zhu.chungangSummary: NURBS method is the standard mathematical method for describing the shapes of curves/surfaces/volumes, and it is extensively used in computer-aided design, computer-aided manufacturing, and computer graphics. The injectivity of NURBS volumes means that they do not have self-intersections. Since the injectivity of parameterizations depend on the signs of their Jacobian functions, and the Jacobian of a NURBS volume is determined by the determinant of its tangent vectors in three directions, we first propose a method to compute the bounding vectors of the tangent cones of NURBS volume in this paper. Then the sufficient condition for the injectivity of NURBS volume is proposed. A checking algorithm is also presented. Some examples are given to verify the effectiveness of the algorithm.An adaptive sampling and domain learning strategy for multivariate function approximation on unknown domainshttps://zbmath.org/1516.650092023-09-19T14:22:37.575876Z"Adcock, Ben"https://zbmath.org/authors/?q=ai:adcock.ben"Cardenas, Juan M."https://zbmath.org/authors/?q=ai:cardenas.juan-m"Dexter, Nick"https://zbmath.org/authors/?q=ai:dexter.nick-cSummary: Many problems arising in computational science and engineering can be described in terms of approximating a smooth function of \(d\) variables, defined over an unknown \textit{domain of interest} \(\Omega\subset\mathbb{R}^d\), from sample data. Here both the underlying dimensionality of the problem (in the case \(d\gg 1\)) as well as the lack of domain knowledge -- with \(\Omega\) potentially irregular and/or disconnected -- are confounding factors for sampling-based methods. Naïve approaches for such problems often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upward of 20\% wasted samples in some problems considered herein. In applications such as surrogate model construction in computational \textit{uncertainty quantification}, the high cost of computing samples necessitates a more efficient sampling procedure. Over the last several years methods for computing such approximations from sample data have been studied in the case of irregular domains, and the advantages of computing sampling measures depending on an approximation space \(P\) of \(\dim (P)=N\) have been shown. More specifically, such approaches confer advantages such as stability and well-conditioning, with an asymptotically optimal sample complexity scaling \(\mathcal{O}(N\log (N))\). The recently proposed \textit{adaptive sampling for general domains} (ASGD) strategy is one such technique to construct these sampling measures. The main contribution of this paper is a procedure to improve upon the ASGD approach by adaptively updating the sampling measure in the case of unknown domains. We achieve this by first introducing a \textit{general domain adaptivity strategy}, which computes an approximation of the function and domain of interest from the sample points. Second, we propose an adaptive sampling strategy, termed \textit{adaptive sampling for unknown domains} (ASUD), which generates sampling measures over a domain that may not be known in advance, based on the ideas introduced in the ASGD approach. We then derive (weighted) \textit{least squares} and \textit{augmented least squares} techniques for polynomial approximation on unknown domains. We present numerical experiments demonstrating the efficacy of the adaptive sampling techniques with \textit{least squares-}based polynomial approximation schemes. Our results show that the ASUD approach consistently achieves errors the same as or smaller than uniform sampling, but using fewer, and often significantly fewer, function evaluations.Guidelines for RBF-FD discretization: numerical experiments on the interplay of a multitude of parameter choiceshttps://zbmath.org/1516.650102023-09-19T14:22:37.575876Z"Le Borne, Sabine"https://zbmath.org/authors/?q=ai:le-borne.sabine"Leinen, Willi"https://zbmath.org/authors/?q=ai:leinen.williSummary: There exist several discretization techniques for the numerical solution of partial differential equations. In addition to classical finite difference, finite element and finite volume techniques, a more recent approach employs radial basis functions to generate differentiation stencils on unstructured point sets. This approach, abbreviated by RBF-FD (radial basis function-finite difference), has gained in popularity since it enjoys several advantages: It is (relatively) straightforward, does not require a mesh and generalizes easily to higher spatial dimensions. However, its application is not quite as blackbox as it may appear at first sight. The computed solution might suffer severely from various sources of errors if RBF-FD parameters are not selected carefully. Through comprehensive numerical experiments, we study the influence of several of these parameters on the condition numbers of intermediate (local) weight matrices, on the condition number of the resulting (global) stiffness matrix and ultimately on the approximation error of the computed discrete solution to the partial differential equation. The parameters of investigation include the type of RBF (and its shape or other parameters if applicable), the degree of polynomial augmentation, the discretization stencil size, the underlying type of point set (structured/unstructured), and the total number of (interior and boundary) points to discretize the PDE, here chosen as a three-dimensional Poisson's problem with Dirichlet boundary conditions. Numerical tests on a sphere as well as tests for the convection-diffusion equation are included in a supplement and demonstrate that the results obtained for the Laplace problem on a cube generalize to wider problem classes. The purpose of this paper is to provide a comprehensive survey on the various components of the basic algorithms for RBF-FD discretization and steer away from potential pitfalls such as computationally more expensive setups which not always lead to more accurate numerical solutions. We guide toward a compatible selection of the multitude of RBF-FD parameters in the basic version of RBF-FD. For many of its components we refer to the literature for more advanced versions.Curves used in highway design and Bézier curveshttps://zbmath.org/1516.650112023-09-19T14:22:37.575876Z"Ayar, Aslı"https://zbmath.org/authors/?q=ai:ayar.asli"Şahin, Bayram"https://zbmath.org/authors/?q=ai:sahin.bayramSummary: Certain curves have been widely used in highway design engineering. The most common and important of these curves is the Bloss curve and it is important to investigate the existence of new Bloss curves for computer-aided geometric design (CAGD). In this paper, we consider certain curves that are very well known in CAGD and check the suitability of such curves in highway design engineering.Refinement for a hybrid boundary representation and its hybrid volume completionhttps://zbmath.org/1516.650122023-09-19T14:22:37.575876Z"Song, Yang"https://zbmath.org/authors/?q=ai:song.yang"Cohen, Elaine"https://zbmath.org/authors/?q=ai:cohen.elaineSummary: With the increasing need for volumetric B-spline representations and the lack of methodologies for creating semi-structured volumetric B-spline representations from B-spline Boundary Representations (B-Rep), hybrid approaches combining semi-structured volumetric B-splines and unstructured Bézier tetrahedra have been introduced, including one that transforms a trimmed B-spline B-Rep first to an untrimmed Hybrid B-Rep (HB-Rep) and then to a Hybrid Volume Representation (HV-Rep). Generally, the effect of \(h\)-\textit{refinement} has not been considered over B-spline hybrid representations. Standard refinement approches to tensor product B-splines and subdivision of Bézier triangles and tetrahedra must be adapted to this representation. In this paper, we analyze possible types of \(h\)-refinement of the HV-Rep. The revised and trim basis functions for HB- and HV-rep depend on a partition of knot intervals. Therefore, a naive \(h\)-refinement can change basis functions in unexpected ways. This paper analyzes the the effect of \(h\)-refinement in reducing error as well. Different \(h\)-refinement strategies are discussed. We demonstrate their differences and compare their respective consequential trade-offs. Recommendations are also made for different use cases.Error estimates for finite differences approximations of the total variationhttps://zbmath.org/1516.650132023-09-19T14:22:37.575876Z"Caillaud, Corentin"https://zbmath.org/authors/?q=ai:caillaud.corentin"Chambolle, Antonin"https://zbmath.org/authors/?q=ai:chambolle.antoninSummary: We present a convergence rate analysis of the Rudin-Osher-Fatemi (ROF) denoising problem for two different discretizations of the total variation. The first is the standard discretization, which induces blurring in some particular diagonal directions. We prove that in a simplified setting corresponding to such a direction, the discrete ROF energy converges to the continuous one with the rate \(h^{2/3}\). The second discretization is based on dual Raviart-Thomas fields and achieves an optimal \(O(h)\) convergence rate for the same quantity, for discontinuous solutions with some standard hypotheses.Two-stage decolorization based on histogram equalization and local variance maximizationhttps://zbmath.org/1516.650142023-09-19T14:22:37.575876Z"Yu, Jing"https://zbmath.org/authors/?q=ai:yu.jing|yu.jing.1"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang.7|li.fang.2|li.fang.1|li.fang.5|li.fang|li.fang.4"Hu, Xuyue"https://zbmath.org/authors/?q=ai:hu.xuyueSummary: Image decolorization is widely used in single-channel image processing, black-and-white printing, etc. Decolorization aims to generate a perceptually satisfactory gray image that preserves the contrast of the color image. It is known that histogram equalization can enhance the global image contrast by effectively spreading out the most frequent intensity values. Meanwhile, local contrast features such as salient edges and local details have large local variances, which can be enhanced by maximizing local variance. Inspired by these facts, we propose a two-stage decolorization method based on histogram equalization and local variance maximization. In the first stage, we assume that the decolorized gray image is a linear combination of the three channels of the color image, and the combination coefficients are three global weights. Then we propose a constrained variational histogram equalization model to optimize the global weights. The resulting gray image has good global contrast. To further enhance the local contrast, in the second stage, we use local weight combination to express the color image and maximize the local variance by forcing the local weights to be close to the global weights. Numerically, the global weights can be estimated by a gradient-based solver or a discrete searching solver, and the local weights are solved by an iterative solver. Theoretically, we discuss the properties of the energy functions and the convergence of the algorithm. Our proposed method better preserves global and local contrast than state-of-the-art decolorization algorithms.Product integration rules by the constrained mock-Chebyshev least squares operatorhttps://zbmath.org/1516.650152023-09-19T14:22:37.575876Z"Dell'Accio, Francesco"https://zbmath.org/authors/?q=ai:dellaccio.francesco"Mezzanotte, Domenico"https://zbmath.org/authors/?q=ai:mezzanotte.domenico"Nudo, Federico"https://zbmath.org/authors/?q=ai:nudo.federico"Occorsio, Donatella"https://zbmath.org/authors/?q=ai:occorsio.donatellaSummary: In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of ``pathological'' behavior, e.g. ``nearly'' singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates.Quadrature processes for efficient calculation of the Clausen functionshttps://zbmath.org/1516.650162023-09-19T14:22:37.575876Z"Golubović, Zora Lj."https://zbmath.org/authors/?q=ai:golubovic.zora-lj"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-vSummary: The Clausen functions arise in numerous applications. An efficient summation/integration method for the numerical calculation of these functions of arbitrary order is proposed in this paper. The method is based on a modification of an earlier method and it does not require the construction of the coefficients in the three-term recurrence relation for the corresponding orthogonal polynomials, but only a transformation of orthogonal polynomials from the real line to the positive semiaxis. Numerical experiments are also included.Efficient CPU and GPU implementations of multicenter integrals over long-range operators using cartesian Gaussian functionshttps://zbmath.org/1516.650172023-09-19T14:22:37.575876Z"Humeniuk, Alexander"https://zbmath.org/authors/?q=ai:humeniuk.alexander"Glover, William J."https://zbmath.org/authors/?q=ai:glover.william-jSummary: We present a library for evaluating multicenter integrals over polarization operators of the form \(x^{m_x}y^{m_y}z^{m_z}r^{- k}C(r)\) using Cartesian Gaussian basis functions. \(m_x, m_y, m_z \geq 0\), \(k > 2\) are integers, while the cutoff function, \(C(r) = (1 - e^{-\alpha r^2})^q\), with \(\alpha\in\mathbb{R}_+\) and certain integer values of \(q\) ensures the existence of the integrals. The formulation developed by \textit{P. Schwerdtfeger} and \textit{H. Silberbach} [Phys. Rev. A (3) 37, No. 8, 2834--2842 (1988; \url{doi:10.1103/PhysRevA.37.2834})] is implemented in an efficient and stable way taking into account a recent fix in one of the equations. A cheap upper bound is presented that allows negligible integrals to be prescreened. The correctness of the analytical integrals was verified by numerical integration. The library provides separate codes for serial CPU and parallel GPU architectures and can be wrapped into a python module.Suboptimality of Gauss-Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothnesshttps://zbmath.org/1516.650182023-09-19T14:22:37.575876Z"Kazashi, Yoshihito"https://zbmath.org/authors/?q=ai:kazashi.yoshihito"Suzuki, Yuya"https://zbmath.org/authors/?q=ai:suzuki.yuya"Goda, Takashi"https://zbmath.org/authors/?q=ai:goda.takashiSummary: The suboptimality of Gauss-Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order \(\alpha\), where the optimality is in the sense of worst-case error. For Gauss-Hermite quadrature, we obtain matching lower and upper bounds, which turn out to be merely of the order \(n^{-\alpha/2}\) with \(n\) function evaluations, although the optimal rate for the best possible linear quadrature is known to be \(n^{-\alpha}\). Our proof of the lower bound exploits the structure of the Gauss-Hermite nodes; the bound is independent of the quadrature weights, and changing the Gauss-Hermite weights cannot improve the rate \(n^{-\alpha/2}\). In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate up to a logarithmic factor.Connections between numerical integration, discrepancy, dispersion, and universal discretizationhttps://zbmath.org/1516.650192023-09-19T14:22:37.575876Z"Temlyakov, Vladimir"https://zbmath.org/authors/?q=ai:temlyakov.vladimir-nSummary: The main goal of this paper is to provide a brief survey of recent results, which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of \textit{fixed volume discrepancy} is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.Fractional Laplacian -- quadrature rules for singular double integrals in 3Dhttps://zbmath.org/1516.650202023-09-19T14:22:37.575876Z"Feist, Bernd"https://zbmath.org/authors/?q=ai:feist.bernd"Bebendorf, Mario"https://zbmath.org/authors/?q=ai:bebendorf.marioSummary: In this article, quadrature rules for the efficient computation of the stiffness matrix for the fractional Laplacian in three dimensions are presented. These rules are based on the Duffy transformation, which is a common tool for singularity removal. Here, this transformation is adapted to the needs of the fractional Laplacian in three dimensions. The integrals resulting from this Duffy transformation are regular integrals over less-dimensional domains. We present bounds on the number of Gauss points to guarantee error estimates which are of the same order of magnitude as the finite element error. The methods presented in this article can easily be adapted to other singular double integrals in three dimensions with algebraic singularities.Expanding the Laurent series with its applicationshttps://zbmath.org/1516.650212023-09-19T14:22:37.575876Z"Adhikari, Ganesh Prasad"https://zbmath.org/authors/?q=ai:adhikari.ganesh-prasadSummary: In Nepal, there are many mathematics subjects taught at university level. Among them, complex analysis is the most powerful. In complex analysis, the Laurent series expansion is a well-known subject because it may be used to find the residues of complex functions around their singularities. It turns out that computing the Laurent series of a function around its singularities is an effective way to calculate the integral of the function along any closed contour around the singularities as well as the residue of the function. Learning the Laurent series concepts can be difficult, and many students struggle to develop adequate understanding, reasoning, and problem-solving skills. Therefore, this article presents multiple practical examples where the Laurent series of a function is found and then utilized to compute the integral of the function over any closed contour around the singularities of the function, based on the theory of the Laurent series.Generalization of the Gauss-Jordan method for solving homogeneous infinite systems of linear algebraic equationshttps://zbmath.org/1516.650222023-09-19T14:22:37.575876Z"Fedorov, F. M."https://zbmath.org/authors/?q=ai:fedorov.f-m"Pavlov, N. N."https://zbmath.org/authors/?q=ai:pavlov.n-n"Potapova, S. V."https://zbmath.org/authors/?q=ai:potapova.s-v"Ivanova, O. F."https://zbmath.org/authors/?q=ai:ivanova.o-f"Shadrin, V. Yu."https://zbmath.org/authors/?q=ai:shadrin.v-yu(no abstract)Mixed precision iterative refinement with sparse approximate inverse preconditioninghttps://zbmath.org/1516.650232023-09-19T14:22:37.575876Z"Carson, Erin"https://zbmath.org/authors/?q=ai:carson.erin"Khan, Noaman"https://zbmath.org/authors/?q=ai:khan.noamanSummary: With the commercial availability of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes have emerged as popular approaches for solving sparse linear systems. Existing analyses of these approaches, however, are based on using full LU factorizations to construct preconditioners for use within GMRES in each refinement step. In practical applications, inexact preconditioning techniques, such as incomplete LU or sparse approximate inverses, are often used for performance reasons. In this work, we investigate the use of sparse approximate inverse preconditioners based on Frobenius norm minimization within GMRES-based iterative refinement. We analyze the computation of sparse approximate inverses in finite precision and derive constraints under which user-specified stopping criteria will be satisfied. We then analyze the behavior of and convergence constraints for a five-precision GMRES-based iterative refinement scheme that uses sparse approximate inverse preconditioning, which we call SPAI-GMRES-IR. Our numerical experiments confirm the theoretical analysis and illustrate the resulting tradeoffs between preconditioner sparsity and GMRES-IR convergence rate.An optimal scheduled learning rate for a randomized Kaczmarz algorithmhttps://zbmath.org/1516.650242023-09-19T14:22:37.575876Z"Marshall, Nicholas F."https://zbmath.org/authors/?q=ai:marshall.nicholas-f"Mickelin, Oscar"https://zbmath.org/authors/?q=ai:mickelin.oscarSummary: We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving \(Ax\approx b+\varepsilon\), where \(Ax=b\) is a consistent linear system and \(\varepsilon\) has independent mean zero random entries. We derive a learning rate schedule which optimizes a bound on the expected error that is sharp in certain cases; in contrast to the exponential convergence of the standard randomized Kaczmarz algorithm, our optimized bound involves the reciprocal of the Lambert-\(W\) function of an exponential.GPMR: an iterative method for unsymmetric partitioned linear systemshttps://zbmath.org/1516.650252023-09-19T14:22:37.575876Z"Montoison, Alexis"https://zbmath.org/authors/?q=ai:montoison.alexis"Orban, Dominique"https://zbmath.org/authors/?q=ai:orban.dominiqueSummary: We introduce an iterative method named \textsc{Gpmr} (general partitioned minimum residual) for solving \(2\times 2\) block unsymmetric linear systems. \textsc{Gpmr} is based on a new process that simultaneously reduces two rectangular matrices to upper Hessenberg form and is closely related to the block-Arnoldi process. \textsc{Gpmr} is tantamount to \textsc{Block-Gmres} with two right-hand sides in which the two approximate solutions are summed at each iteration, but its storage and work per iteration are similar to those of \textsc{Gmres}. We compare the performance of \textsc{Gpmr} with \textsc{Gmres} on linear systems from the SuiteSparse Matrix Collection. In our experiments, \textsc{Gpmr} terminates significantly earlier than \textsc{Gmres} on a residual-based stopping condition with an improvement ranging from around 10\% up to 50\% in terms of number of iterations.A deterministic Kaczmarz algorithm for solving linear systemshttps://zbmath.org/1516.650262023-09-19T14:22:37.575876Z"Shao, Changpeng"https://zbmath.org/authors/?q=ai:shao.changpengSummary: We propose a new deterministic Kaczmarz algorithm for solving consistent linear systems \(A\mathbf{x}=\mathbf{b}\). Basically, the algorithm replaces orthogonal projections with reflections in the original scheme of Stefan Kaczmarz. Building on this, we give a geometric description of solutions of linear systems. Suppose \(A\) is \(m\times n\), we show that the algorithm generates a series of points distributed with patterns on an \((n-1)\)-sphere centered on a solution. These points lie evenly on \(2m\) lower-dimensional spheres \(\{\mathbb{S}_{k0},\mathbb{S}_{k1}\}_{k=1}^m\), with the property that for any \(k\), the midpoint of the centers of \(\mathbb{S}_{k0},\mathbb{S}_{k1}\) is exactly a solution of \(A\mathbf{x}=\mathbf{b}\). With this discovery, we prove that taking the average of \(O(\eta (A)\log (1/\varepsilon))\) points on any \(\mathbb{S}_{k0}\cup\mathbb{S}_{k1}\) effectively approximates a solution up to relative error \(\varepsilon \), where \(\eta (A)\) characterizes the eigengap of the orthogonal matrix produced by the product of \(m\) reflections generated by the rows of \(A\). We also analyze the connection between \(\eta(A)\) and \(\kappa(A)\), the condition number of \(A\). In the worst case \(\eta (A)=O(\kappa^2(A)\log m)\), while for random matrices \(\eta (A)=O(\kappa (A))\) on average. Finally, we prove that the algorithm indeed solves the linear system \(A^{\mathtt{T}}W^{-1}A\mathbf{x}=A^{\mathtt{T}}W^{-1}\mathbf{b}\), where \(W\) is the lower-triangular matrix such that \(W+W^{\mathtt{T}}=2AA^{\mathtt{T}}\). The connection between this linear system and the original one is studied. The numerical tests indicate that this new Kaczmarz algorithm has comparable performance to randomized (block) Kaczmarz algorithms.A randomised iterative method for solving factorised linear systemshttps://zbmath.org/1516.650272023-09-19T14:22:37.575876Z"Zhao, Jing"https://zbmath.org/authors/?q=ai:zhao.jing|zhao.jing.1|zhao.jing.2|zhao.jing.3|zhao.jing.4"Wang, Xiang"https://zbmath.org/authors/?q=ai:wang.xiang.2|wang.xiang"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua.1|zhang.jianhuaSummary: For solving a linear system with a large-scale coefficient matrix which is stored in a factorised form, we present a new interlaced randomised iterative method. The new method can take advantage of the factored form and avoid performing matrix multiplications. Furthermore, its convergence property is studied and theoretical analysis reveals that it is linearly convergent. Finally, some numerical results are given to confirm the effectiveness of the proposed method.Global convergence of triangularized orthogonalization-free methodhttps://zbmath.org/1516.650282023-09-19T14:22:37.575876Z"Gao, Weiguo"https://zbmath.org/authors/?q=ai:gao.weiguo"Li, Yingzhou"https://zbmath.org/authors/?q=ai:li.yingzhou"Lu, Bichen"https://zbmath.org/authors/?q=ai:lu.bichenSummary: This paper proves the global convergence of a triangularized orthogonalization-free method (TriOFM). TriOFM, in general, applies a triangularization idea to the gradient of an objective function and removes the rotation invariance in minimizers. More precisely, in this paper, the TriOFM works as an eigensolver for sizeable sparse matrices and obtains eigenvectors without any orthogonalization step. Due to the triangularization, the iteration is a discrete-time flow in a non-conservative vector field. The global convergence relies on the stable manifold theorem, whereas the convergence to stationary points is proved in detail in this paper. We provide two proofs inspired by the noisy power method and the noisy optimization method, respectively.The joint bidiagonalization method for large GSVD computations in finite precisionhttps://zbmath.org/1516.650292023-09-19T14:22:37.575876Z"Jia, Zhongxiao"https://zbmath.org/authors/?q=ai:jia.zhongxiao"Li, Haibo"https://zbmath.org/authors/?q=ai:li.haiboSummary: The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair \(\{A,L\}\). We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of \(\{A,L\}\). The results show that, under some mild conditions, the semiorthogonality of Lanczos-type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.Matrix-eigenvalue method for the quantum harmonic oscillatorhttps://zbmath.org/1516.650302023-09-19T14:22:37.575876Z"Rhee, Noah"https://zbmath.org/authors/?q=ai:rhee.noah-h"Leibsle, Fred M."https://zbmath.org/authors/?q=ai:leibsle.fred-mSummary: In this short note we discuss how to solve the quantum harmonic oscillator. There is a well known traditional method of solving the quantum harmonic oscillator. Here we present a matrix-eigenvalue method. We also present another equivalent matrix-eigenvalue method, which has pedagogical value.Krylov subspace approach to core problems within multilinear approximation problems: a unifying frameworkhttps://zbmath.org/1516.650312023-09-19T14:22:37.575876Z"Hnětynková, Iveta"https://zbmath.org/authors/?q=ai:hnetynkova.iveta"Plešinger, Martin"https://zbmath.org/authors/?q=ai:plesinger.martin"Žáková, Jana"https://zbmath.org/authors/?q=ai:zakova.janaSummary: Error contaminated linear approximation problems appear in a large variety of applications. The presence of redundant or irrelevant data complicates their solution. It was shown that such data can be removed by the core reduction yielding a minimally dimensioned subproblem called the core problem. Direct (SVD or Tucker decomposion-based) reduction has been introduced previously for problems with matrix models and vector, or matrix, or tensor observations; and also for problems with bilinear models. For the cases of vector and matrix observations a Krylov subspace method, the generalized Golub-Kahan bidiagonalization, can be used to extract the core problem. In this paper, we first unify previously studied variants of linear approximation problems under the general framework of a multilinear approximation problem. We show how the direct core reduction can be extended to it. Then we show that the generalized Golub-Kahan bidiagonalization yields the core problem for any multilinear approximation problem. This further allows one to prove various properties of core problems, in particular, we give upper bounds on the multiplicity of singular values of reduced matrices.Cyclic gradient based iterative algorithm for a class of generalized coupled Sylvester-conjugate matrix equationshttps://zbmath.org/1516.650322023-09-19T14:22:37.575876Z"Wang, Wenli"https://zbmath.org/authors/?q=ai:wang.wenli"Qu, Gangrong"https://zbmath.org/authors/?q=ai:qu.gangrong"Song, Caiqin"https://zbmath.org/authors/?q=ai:song.caiqinSummary: This paper focuses on the numerical solution of a class of generalized coupled Sylvester-conjugate matrix equations, which are general and contain many significance matrix equations as special cases, such as coupled discrete-time/continuous-time Markovian jump Lyapunov matrix equations, stochastic Lyapunov matrix equation, etc. By introducing the modular operator, a cyclic gradient based iterative (CGI) algorithm is provided. Different from some previous iterative algorithms, the most significant improvement of the proposed algorithm is that less information is used during each iteration update, which is conducive to saving memory and improving efficiency. The convergence of the proposed algorithm is discussed, and it is verified that the algorithm converges for any initial matrices under certain assumptions. Finally, the effectiveness and superiority of the proposed algorithm are verified with some numerical examples.Structured gradient descent for fast robust low-rank Hankel matrix completionhttps://zbmath.org/1516.650332023-09-19T14:22:37.575876Z"Cai, Hanqin"https://zbmath.org/authors/?q=ai:cai.hanqin"Cai, Jian-Feng"https://zbmath.org/authors/?q=ai:cai.jian-feng"You, Juntao"https://zbmath.org/authors/?q=ai:you.juntaoSummary: We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from partial observation. In this paper, we explore the convenient Hankel structure and propose a novel nonconvex algorithm, coined Hankel structured gradient descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing- and sample-efficient compared to the state of the art. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.Properties of the solution set of absolute value equations and the related matrix classeshttps://zbmath.org/1516.650342023-09-19T14:22:37.575876Z"Hladík, Milan"https://zbmath.org/authors/?q=ai:hladik.milanSummary: The absolute value equations (AVE) problem is an algebraic problem of solving \(Ax+|x|=b\). So far, most of the research has focused on methods for solving AVE, but we address the problem itself by analyzing properties of AVE and the corresponding solution set. In particular, we investigate topological properties of the solution set, such as convexity, boundedness, or connectedness, or whether it consists of finitely many solutions. Further, we address problems related to the nonnegativity of solutions such as solvability or unique solvability. AVE can be formulated by means of different optimization problems, and in this regard we are interested in how the solutions of AVE are related with optima, Karush-Kuhn-Tucker points, and feasible solutions of these optimization problems. We characterize the matrix classes associated with the above mentioned properties and inspect the computational complexity of the recognition problem; some of the classes are polynomially recognizable, but some others are proved to be NP-hard. For the intractable cases, we propose various sufficient conditions. We also post new challenging problems that were raised during the investigation of the problem.The transport of images method: computing all zeros of harmonic mappings by continuationhttps://zbmath.org/1516.650352023-09-19T14:22:37.575876Z"Sète, Olivier"https://zbmath.org/authors/?q=ai:sete.olivier"Zur, Jan"https://zbmath.org/authors/?q=ai:zur.janSummary: We present a continuation method to compute all zeros of a harmonic mapping \(f\) in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solutions of \(f(z) = \eta\) with \(|\eta|\) sufficiently large and then track all solutions as \(\eta\) tends to 0 to finally obtain all zeros of \(f\). Using theoretical results on harmonic mappings we analyze where and how the number of solutions of \(f(z) = \eta\) changes and incorporate this into the method. We prove that our method is guaranteed to compute all zeros, as long as none of them is singular. In our numerical examples the method always terminates with the correct number of zeros, is very fast compared to general purpose root finders and is highly accurate in terms of the residual. An easy-to-use MATLAB implementation is freely available online.A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetryhttps://zbmath.org/1516.650362023-09-19T14:22:37.575876Z"Appeltans, Pieter"https://zbmath.org/authors/?q=ai:appeltans.pieter"Michiels, Wim"https://zbmath.org/authors/?q=ai:michiels.wimSummary: This work considers a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axes. We propose a method to iteratively approximate the eigenvalues closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end, our method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a ``real skew-Hamiltonian''-like structure. Furthermore, it is proven that the Krylov subspace constructed by applying this operator satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that, even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [\textit{V. Mehrmann} and \textit{D. Watkins}, SIAM J. Sci. Comput. 22, No. 6, 1905--1925 (2001; Zbl 0986.65033)] to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite-dimensional linear algebra operations. The performance of the resulting numerical algorithm is verified for two example problems: the first example illustrates the advantage of the proposed approach in preserving purely imaginary eigenvalues when working in finite precision, while the second one demonstrates its applicability to a large scale problem.A smallest singular value method for nonlinear eigenvalue problemshttps://zbmath.org/1516.650372023-09-19T14:22:37.575876Z"Sadkane, Miloud"https://zbmath.org/authors/?q=ai:sadkane.miloudSummary: A Newton-type method for the eigenvalue problem of analytic matrix functions is described and analysed. The method finds the eigenvalue and eigenvector, respectively, as a point in the level set of the smallest singular value function and the corresponding right singular vector. The algorithmic aspects are discussed and illustrated by numerical examples.The role of a priori estimates in the method of non-local continuation of solution by parameterhttps://zbmath.org/1516.650382023-09-19T14:22:37.575876Z"Sidorov, Nikolaĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:sidorov.nikolai-aleksandrovichSummary: An iterative method for continuation of solutions with respect to a parameter is proposed. The nonlocal case is studied when the parameter belongs to the segment of the real axis. An iterative scheme for continuing the solution is constructed for a linear equation in Banach spaces with a linear operator continuously depending on the parameter, satisfying the Lipschitz condition with respect to the parameter. The generalization of this result on a nonlinear equation in Banach spaces is proposed. The iterative scheme of the method of continuation of the solution by parameter using the Newton-Kantorovich method is constructed. An priori estimates of solutions enable solution construction for arbitrary parameters.An inverse free Broyden's method for solving equationshttps://zbmath.org/1516.650392023-09-19T14:22:37.575876Z"Argyros, Ioannis K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"George, Santhosh"https://zbmath.org/authors/?q=ai:george.santhoshSummary: Based on a center-Lipschitz-type condition and our idea of the restricted convergence domain, we present a new semi-local convergence analysis for an inverse free Broyden's method (BM) in order to approximate a locally unique solution of an equation in a Hilbert space setting. The operators involved have regularly continuous divided differences. This way we provide weaker sufficient semi-local convergence conditions, tighter error bounds, and a more precise information on the location of the solution. Hence, our approach extends the applicability of BM under the same hypotheses as before. Finally, we consider some special cases.Extended local convergence and comparisons for two three-step Jarratt-type methods under the same conditionshttps://zbmath.org/1516.650402023-09-19T14:22:37.575876Z"Argyros, Ioannis K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"George, Santhosh"https://zbmath.org/authors/?q=ai:george.santhosh"Argyros, Christopher"https://zbmath.org/authors/?q=ai:argyros.christopher-ioannisSummary: We extend and compare two three-step Jarratt-type methods for solving a nonlinear equation under the same conditions. Our convergence analysis is based on the first Fréchet derivative that only appears in the method. Numerical examples where the theoretical results are tested complete the paper.Extended convergence ball for an efficient eighth order method using only the first derivativehttps://zbmath.org/1516.650412023-09-19T14:22:37.575876Z"Argyros, Ioannis K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"Sharma, Debasis"https://zbmath.org/authors/?q=ai:sharma.debasis"Argyros, Christopher I."https://zbmath.org/authors/?q=ai:argyros.christopher-ioannis"Parhi, Sanjaya Kumar"https://zbmath.org/authors/?q=ai:parhi.sanjaya-kumar"Sunanda, Shanta Kumari"https://zbmath.org/authors/?q=ai:sunanda.shanta-kumariSummary: We develop an extended convergence ball for an efficient eighth order method to obtain numerical solutions of Banach space valued nonlinear models. Convergence of this algorithm has previously been shown using assumptions up to the ninth derivative. However, in our convergence theorem, we use only the first derivative. As a consequence, in contrast to previous ideas, the results on calculable error bounds, convergence radius and uniqueness zone for the solution are provided. Furthermore, this scheme is applied to several complex polynomials and related attraction basins are displayed. The results of numerical tests are presented and compared with the earlier technique. We arrive at the conclusion that the suggested analysis produces much larger convergence radii in all tests. Hence, we expand the convergence domain of this iterative formula.Data-consistent neural networks for solving nonlinear inverse problemshttps://zbmath.org/1516.650422023-09-19T14:22:37.575876Z"Boink, Yoeri E."https://zbmath.org/authors/?q=ai:boink.yoeri-e"Haltmeier, Markus"https://zbmath.org/authors/?q=ai:haltmeier.markus"Holman, Sean"https://zbmath.org/authors/?q=ai:holman.sean-f"Schwab, Johannes"https://zbmath.org/authors/?q=ai:schwab.johannesSummary: Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve its solution. Empirical evidence shows that plain two-step methods provide high-quality reconstructions, but they lack a convergence analysis as known for classical regularization methods. In this paper we formalize the use of such two-step approaches in the context of classical regularization theory. We propose data-consistent neural networks that can be combined with classical regularization methods. This yields a data-driven regularization method for which we provide a convergence analysis with respect to noise. Numerical simulations show that compared to standard two-step deep learning methods, our approach provides better stability with respect to out of distribution examples in the test set, while performing similarly on test data drawn from the distribution of the training set. Our method provides a stable solution approach to inverse problems that beneficially combines the known nonlinear forward model with available information on the desired solution manifold in training data.A fast data-driven iteratively regularized method with convex penalty for solving ill-posed problemshttps://zbmath.org/1516.650432023-09-19T14:22:37.575876Z"Gao, Guangyu"https://zbmath.org/authors/?q=ai:gao.guangyu"Han, Bo"https://zbmath.org/authors/?q=ai:han.bo"Fu, Zhenwu"https://zbmath.org/authors/?q=ai:fu.zhenwu"Tong, Shanshan"https://zbmath.org/authors/?q=ai:tong.shanshanSummary: We propose a new iterative regularization method for solving inverse problems in Hilbert spaces. The iterative process of the proposed method combines classical iterative regularization format and Data-Driven approach. Data-Driven technique is based on the idea of deep learning to estimate the interior of a black box through a training set, so as to solve problems better and faster in some cases. In order to capture the special feature of solutions, convex functions are utilized to be penalty terms. Algorithmically, the two-point gradient acceleration strategy based on homotopy perturbation method is applied to the iterative scheme, which makes the method have satisfactory acceleration effect. We provide convergence analysis of the method under standard assumptions for iterative regularization methods. Finally, several numerical experiments are presented to show the effectiveness and acceleration effect of our method.Globally convergent Dai-Liao conjugate gradient method using quasi-Newton update for unconstrained optimizationhttps://zbmath.org/1516.650442023-09-19T14:22:37.575876Z"Chen, Yuting"https://zbmath.org/authors/?q=ai:chen.yutingSummary: Using quasi-Newton update and acceleration scheme, a new Dai-Liao conjugate gradient method that does not need computing or storing any approximate Hessian matrix of the objective function is developed for unconstrained optimization. It is shown that the search direction derived from a modified Perry matrix not only possesses sufficient descent condition but also fulfills Dai-Liao conjugacy condition at each iteration. Under certain assumptions, we establish the global convergence of the proposed method for uniformly convex function and general function, respectively. The numerical results illustrate that the presented method can effectively improve the numerical performance and successfully solve the test problems with a maximum dimension of \(100000\).A derivative-free conjugate gradient method for large-scale nonlinear systems of monotone equationshttps://zbmath.org/1516.650452023-09-19T14:22:37.575876Z"Gao, Jing"https://zbmath.org/authors/?q=ai:gao.jing"Li, Yanran"https://zbmath.org/authors/?q=ai:li.yanran"Cao, Mingyuan"https://zbmath.org/authors/?q=ai:cao.mingyuan"Yang, Yueting"https://zbmath.org/authors/?q=ai:yang.yueting"Bai, Xue"https://zbmath.org/authors/?q=ai:bai.xueSummary: This paper presents a derivative-free conjugate gradient type algorithm for large-scale nonlinear systems of monotone equations. New search directions with superior numerical performance are constructed by introducing a new conjugate parameter and particular spectral parameters. These search directions inherit the numerical stability of RMIL search direction and satisfy the sufficient descent condition independent of step size. The method combines the hyperplane projection and the derivative-free line search technique to compute the iteration points. Under some appropriate assumptions, the global convergence of the given methods is established. Numerical experiments indicate that the proposed algorithms are effective.General Hölder smooth convergence rates follow from specialized rates assuming growth boundshttps://zbmath.org/1516.650462023-09-19T14:22:37.575876Z"Grimmer, Benjamin"https://zbmath.org/authors/?q=ai:grimmer.benjaminSummary: Often in the analysis of first-order methods for both smooth and nonsmooth optimization, assuming the existence of a growth/error bound or KL condition facilitates much stronger convergence analysis. Hence, separate analysis is typically needed for the general case and for the growth bounded cases. We give meta-theorems for deriving general convergence rates from those assuming a growth lower bound. Applying this simple but conceptually powerful tool to the proximal point, subgradient, bundle, dual averaging, gradient descent, Frank-Wolfe and universal accelerated methods immediately recovers their known convergence rates for general convex optimization problems from their specialized rates. New convergence results follow for bundle methods, dual averaging and Frank-Wolfe. Our results can lift any rate based on Hölder continuous gradients and Hölder growth bounds. Moreover, our theory provides simple proofs of optimal convergence lower bounds under Hölder growth from textbook examples without growth bounds.On the inertial relaxed CQ algorithm in Hilbert spaceshttps://zbmath.org/1516.650472023-09-19T14:22:37.575876Z"Li, Haiying"https://zbmath.org/authors/?q=ai:li.haiying"Wang, Fenghui"https://zbmath.org/authors/?q=ai:wang.fenghui"Yu, Hai"https://zbmath.org/authors/?q=ai:yu.haiSummary: In this paper, we study the inertial relaxed CQ algorithm for solving a split feasibility problem in Hilbert spaces. For this algorithm, we establish two convergence theorems under two different conditions. The first condition is weaker than the existing condition, and the second condition is completely different from the existing one. Moreover, the preliminary numerical experiment indicates that our proposed algorithms converge faster than the existing algorithms.Apodizer design to efficiently couple light into a fiber Bragg gratinghttps://zbmath.org/1516.650482023-09-19T14:22:37.575876Z"Adriazola, Jimmie"https://zbmath.org/authors/?q=ai:adriazola.jimmie"Goodman, Roy H."https://zbmath.org/authors/?q=ai:goodman.roy-hSummary: We provide an optimal control framework for efficiently coupling light in a bare fiber into Bragg gratings with cubic nonlinearity. The light-grating interaction excites gap solitons, a type of localized nonlinear coherent state which propagates with a central frequency in the forbidden band gap, resulting in a dramatically slower group velocity. Due to the nature of the band gap, a substantial amount of light is back-reflected by the grating's strong reflective properties. We optimize, via a projected gradient descent method, the transmission efficiency of previously designed nonuniform grating structures in order to couple more slow light into the grating. We further explore the space of possible grating designs, using genetic algorithms, along with a previously unexplored design parameter: the grating chirp. Through these methods, we find structures that couple a greater fraction of light into the grating with the added bonus of creating slower pulses.Numerical analysis of a general elliptic variational-hemivariational inequalityhttps://zbmath.org/1516.650492023-09-19T14:22:37.575876Z"Han, Weimin"https://zbmath.org/authors/?q=ai:han.weimin"Sofonea, Mircea"https://zbmath.org/authors/?q=ai:sofonea.mirceaSummary: This paper is devoted to the numerical analysis of a general elliptic variational-hemivariational inequality. After a review of a solution existence and uniqueness result, we introduce a family of Galerkin methods to solve the problem. We prove the convergence of the numerical method under the minimal solution regularity condition available from the existence result and derive a Céa's inequality for error estimation of the numerical solutions. Then, we apply the results for the numerical analysis of a variational-hemivariational inequality in the study of a static problem which models the contact of an elastic body with a reactive foundation. In particular, under appropriate solution regularity conditions, we derive an optimal order error estimate for the linear finite element solution.An inertial modified algorithm for solving variational inequalitieshttps://zbmath.org/1516.650502023-09-19T14:22:37.575876Z"Hieu, Dang Van"https://zbmath.org/authors/?q=ai:dang-van-hieu."Quy, Pham Kim"https://zbmath.org/authors/?q=ai:quy.pham-kimSummary: The paper deals with an inertial-like algorithm for solving a class of variational inequality problems involving Lipschitz continuous and strongly pseudomonotone operators in Hilbert spaces. The presented algorithm can be considered a combination of the modified subgradient extragradient-like algorithm and inertial effects. This is intended to speed up the convergence properties of the algorithm. The main feature of the new algorithm is that it is done without the prior knowledge of the Lipschitz constant and the modulus of strong pseudomonotonicity of the cost operator. Several experiments are performed to illustrate the convergence and computational performance of the new algorithm in comparison with others having similar features. The numerical results have confirmed that the proposed algorithm has a competitive advantage over the existing methods.The application of numerical methods to solve linear systems with a time delayhttps://zbmath.org/1516.650512023-09-19T14:22:37.575876Z"Grebenshchikov, Boris Georgievich"https://zbmath.org/authors/?q=ai:grebenshchikov.boris-georgievich"Zagrebina, Sof'ya Aleksandrovna"https://zbmath.org/authors/?q=ai:zagrebina.sofya-aleksandrovna"Lozhnikov, Andreĭ Borisovich"https://zbmath.org/authors/?q=ai:lozhnikov.andrey-borisovichSummary: This paper considers the application of modified numerical methods for solving differential equations with a delay which linearly depends on time. Since the delay increases indefinitely, it is also necessary to apply asymptotic methods to analyze the behavior of the solutions of such systems. The paper establishes the asymptotic properties of the systems under study, which significantly affect the accuracy of the numerical calculation. Given the unbounded delay and the instability of the solutions and to clarify the properties of the solution of such systems, it is useful to know the asymptotic properties of the derivatives having an order greater than one. Under the conditions formulated in the article, these derivatives tend to zero as \(t\to \infty \). This property makes it possible to apply finite-order numerical methods (such as the Runge-Kutta method and the modified Euler method). As an illustration of the effectiveness of the methods developed, the article calculates the vertical oscillations of a locomotive pantograph moving at a constant speed when interacting with the contact wire. The numerical methods allow the study of the asymptotic behavior of more complex systems containing both constant and linear delay. Note that the use of numerical methods for calculating the solution reveal the instability of the solution of the systems under study and can be used to stabilize some systems containing an unlimited (not necessarily linear) delay.Phase error analysis of implicit Runge-Kutta methods: new classes of minimal dissipation low dispersion high order schemeshttps://zbmath.org/1516.650522023-09-19T14:22:37.575876Z"Giri, Subhajit"https://zbmath.org/authors/?q=ai:giri.subhajit"Sen, Shuvam"https://zbmath.org/authors/?q=ai:sen.shuvamSummary: In the current research, we analyze dissipation and dispersion characteristics of the most accurate two and three-stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their A-stability and high stage order, are observed to carry minimum dissipation error along with the highest possible dispersive order in their respective classes. Investigation reveals that these schemes are inherently optimized to carry low phase error only at small wavenumber. It is noticed that a unique scheme, although usually sought, might not be best across diverse temporal step sizes. As larger temporal step size is imperative in conjunction with implicit R-K methods for physical problems, we thoroughly investigate to derive a class of minimum dissipation and optimally low dispersion implicit R-K schemes. Schemes are obtained by cutting down amplification error and maximum reduction of weighted phase error, suggest better accuracy for relatively bigger and varied CFL numbers. A potentially generalizable algorithm is used to design stable implicit R-K methods. As the work focuses on two and three-stage schemes, a comprehensive comparison using numerical test cases document only modest gain in accuracy or efficiency over Gauss methods.A new optimality property of Strang's splittinghttps://zbmath.org/1516.650532023-09-19T14:22:37.575876Z"Casas, Fernando"https://zbmath.org/authors/?q=ai:casas.fernando"Sanz-Serna, Jesús María"https://zbmath.org/authors/?q=ai:sanz-serna.jesus-maria"Shaw, Luke"https://zbmath.org/authors/?q=ai:shaw.lukeSummary: For systems of the form \(\dot q=M^{-1} p\), \(\dot p=-Aq+f(q)\), common in many applications, we analyze splitting integrators based on the (linear/nonlinear) split systems \(\dot q=M^{-1} p\), \(\dot p=-Aq\) and \(\dot q=0\), \(\dot p=f(q)\). We show that the well-known Strang splitting is optimally stable in the sense that, when applied to a relevant model problem, it has a larger stability region than alternative integrators. This generalizes a well-known property of the common Störmer/Verlet/leapfrog algorithm, which of course arises from Strang splitting based on the (kinetic/potential) split systems \(\dot q=M^{-1} p\), \(\dot p=0\) and \(\dot q=0\), \(\dot p=-Aq+f(q)\).Corrigendum to: ``Bratu's problem: a novel approach using fixed-point iterations and Green's functions''https://zbmath.org/1516.650542023-09-19T14:22:37.575876ZFrom the text: It has been brought to our attention by Professor Shih-Hsiang Chang, Far East University, Taiwan that there are sign errors in some of the equations presented in the article [\textit{H. Q. Kafri} and \textit{S. A. Khuri}, ibid. 198, 97--104 (2016; Zbl 1344.65065)]. These errors have been confirmed by the authors. Specifically, there is a sign error in the calculation of the Green's function, such that Eq. (2.21) is in error. Consequently, the sign of the integral in the operator \(K[u]\), defined in Eq. (2.22), should be changed from positive to negative. The Green's function presented in (3.36) derived from (2.21) should also have its sign changed. Nevertheless, the final iterative algorithm (3.37) is correct.Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domainshttps://zbmath.org/1516.650552023-09-19T14:22:37.575876Z"Izadi, Mohammad"https://zbmath.org/authors/?q=ai:izadi.mohammad-a|izadi.mohammad"Yüzbaşı, Şuayip"https://zbmath.org/authors/?q=ai:yuzbasi.suayip"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carloThe authors develop an effective matrix method based upon generalized Bessel polynomials to find an approximate solution of fractional-order Bagley-Torvik differential equations over a long time domain. Representing all unknowns in the matrix forms and with the help of collocation points, the original equations lead to an algebraic system of linear equations. The error and convergence results on the spectral Bessel solutions are investigated. In particular, a fast algorithm with linear complexity for computing the \(q\)-th order fractional derivative of the basis functions in the vectorized form is presented. Numerical experiments are presented to show the efficiency of the approach suggested by the authors.
Reviewer: Abdallah Bradji (Annaba)On one method for solving of a static beam problemhttps://zbmath.org/1516.650562023-09-19T14:22:37.575876Z"Kachakhidze, N."https://zbmath.org/authors/?q=ai:kachakhidze.nikoloz"Peradze, J."https://zbmath.org/authors/?q=ai:peradze.jemal"Tsiklauri, Z."https://zbmath.org/authors/?q=ai:tsiklauri.zviad-iSummary: In the article, the variational-iterative method is used to solve a boundary value problem that describes the static state of a beam. The error of the method is estimated and its effectiveness is checked by an example.On the generality of methods of mathematical physics and numerical analysis for some boundary value problems. Ihttps://zbmath.org/1516.650572023-09-19T14:22:37.575876Z"Vashakmadzel, T. S."https://zbmath.org/authors/?q=ai:vashakmadzel.t-s"Gülve, Y. F."https://zbmath.org/authors/?q=ai:gulve.y-fSummary: We consider the problems of creation the convergence difference schemes and numerical realization for the estimate of order of arithmetic operations needed for finding an approximate solution of Dirichlet problem for biharmonic equation in multidimensional cube. In this paper we consider and analyse one-dim case.Multiresolution method for bending of plates with complex shapeshttps://zbmath.org/1516.650582023-09-19T14:22:37.575876Z"Wang, Jizeng"https://zbmath.org/authors/?q=ai:wang.jizeng"Feng, Yonggu"https://zbmath.org/authors/?q=ai:feng.yonggu"Xu, Cong"https://zbmath.org/authors/?q=ai:xu.cong"Liu, Xiaojing"https://zbmath.org/authors/?q=ai:liu.xiaojing"Zhou, Youhe"https://zbmath.org/authors/?q=ai:zhou.youheSummary: A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains. To realize this method, we design a new wavelet basis function, by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains. In the solution of differential equations, various derivatives of the unknown function are denoted as new functions. Then, the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals. Therefore, the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative. During the application of the proposed method, boundary conditions can be automatically included in the integration operations, and relevant matrices can be assured to exhibit perfect sparse patterns. As examples, we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes. By comparing the solutions obtained by the proposed method with the exact solutions, the new multiresolution method is found to have a convergence rate of fifth order. The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method (FEM) with tens of thousands of elements. In addition, because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order, we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.Numerical method for solving fractional Sturm-Liouville eigenvalue problems of order two using Genocchi polynomialshttps://zbmath.org/1516.650592023-09-19T14:22:37.575876Z"Aghazadeh, A."https://zbmath.org/authors/?q=ai:aghazadeh.a"Mahmoudi, Y."https://zbmath.org/authors/?q=ai:mahmoudi.yaghoub"Dastmalchi, Saei F."https://zbmath.org/authors/?q=ai:dastmalchi.saei-farhadSummary: A new numerical scheme based on Genocchi polynomials is constructed
to solve fractional Sturm-Liouville problems of order two in which the fractional derivative is considered in the Caputo sense. First, the differential equation with boundary conditions is converted into the corresponding integral equation form. Next, the fractional integration and derivation operational matrices for Genocchi polynomials, are introduced and applied
for approximating the eigenvalues of the problem. Then, the proposed polynomials are applied to approximate the corresponding eigenfunctions. Finally, some examples are presented to illustrate the efficiency and accuracy of the numerical method. The results show that the proposed method is better than some other approximations involving orthogonal bases.Stability domains of explicit multistep methodshttps://zbmath.org/1516.650602023-09-19T14:22:37.575876Z"Kireev, I. V."https://zbmath.org/authors/?q=ai:kireev.igor-v"Novikov, A. E."https://zbmath.org/authors/?q=ai:novikov.anton-evgenyevich"Novikov, E. A."https://zbmath.org/authors/?q=ai:novikov.evgeny-a(no abstract)Parallel numerical continuation of periodic responses of local nonlinear systemshttps://zbmath.org/1516.650612023-09-19T14:22:37.575876Z"Wang, Qian"https://zbmath.org/authors/?q=ai:wang.qian.10|wang.q-jane|wang.qian.3"Liu, Yi"https://zbmath.org/authors/?q=ai:liu.yi|liu.yi.3"Liu, Heng"https://zbmath.org/authors/?q=ai:liu.heng"Fan, Hongwei"https://zbmath.org/authors/?q=ai:fan.hongwei"Jing, Minqing"https://zbmath.org/authors/?q=ai:jing.minqing(no abstract)High-order semi-discrete central-upwind schemes with Lax-Wendroff-type time discretizations for Hamilton-Jacobi equationshttps://zbmath.org/1516.650622023-09-19T14:22:37.575876Z"Abedian, Rooholah"https://zbmath.org/authors/?q=ai:abedian.rooholahSummary: A new fifth-order, semi-discrete central-upwind scheme with a Lax-Wendroff time discretization procedure for solving Hamilton-Jacobi (HJ) equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. Unlike most of the commonly used high-order upwind schemes, the new scheme is formulated as a Godunov-type method. The new scheme is based on the flux Kurganov, Noelle and Petrova (KNP flux). The spatial discretization is based on a symmetrical weighted essentially non-oscillatory reconstruction of the derivative. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. Various numerical experiments are performed to demonstrate the accuracy and stability properties of the new method. As a result, comparing with other fifth-order schemes for HJ equations, the major advantage of the new scheme is more cost effective for certain problems while the new method exhibits smaller errors without any increase in the complexity of the computations.High-order accurate methods based on difference potentials for 2D parabolic interface modelshttps://zbmath.org/1516.650632023-09-19T14:22:37.575876Z"Albright, Jason"https://zbmath.org/authors/?q=ai:albright.jason"Epshteyn, Yekaterina"https://zbmath.org/authors/?q=ai:epshteyn.yekaterina"Xia, Qing"https://zbmath.org/authors/?q=ai:xia.qingSummary: Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are essential for the resolution of a wide range of temporal and spatial scales in many partial differential equations based models from Biology, Materials Science and Physics. In this paper we continue our work [\textit{J. Albright} et al., Appl. Numer. Math. 93, 87--106 (2015; Zbl 1326.65103)] started in 1D, and we develop high-order accurate methods based on the Difference Potentials for 2D parabolic interface/composite domain problems. Extensive numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes.Transient swelling-induced finite bending of hydrogel-based bilayers: analytical and FEM approacheshttps://zbmath.org/1516.650642023-09-19T14:22:37.575876Z"Amiri, A."https://zbmath.org/authors/?q=ai:amiri.amirhossein|amiri.ashkan|amiri.amir-mohammad|amiri.azita|amiri.alireza-shamekhi|amiri.aboutaleb|amiri.akbar|amiri.a-r|amiri.ali|amiri.aboubacar|amiri.arij|amiri.afshin|amiri.aryan|amiri.abdolreza"Baniassadi, M."https://zbmath.org/authors/?q=ai:baniassadi.majid"Baghani, M."https://zbmath.org/authors/?q=ai:baghani.mostafaSummary: Hydrogels with their time-dependent intrinsic behaviors have recently been used widely in soft structures as sensors/actuators. One of the most interesting structures is the bilayer made up of hydrogels which may undergo swelling-induced bending. In this work, by proposing a semi-analytical method, the transient bending of hydrogel-based bilayers is investigated. Utilizing nonlinear solid mechanics, a robust semi-analytical solution is developed which captures the transient finite bending of hydrogel-based bilayers. Moreover, the multiphysics model of the hydrogels is implemented in the finite element method (FEM) framework to verify the developed semi-analytical procedure results. The effects of different material properties are investigated to illustrate the nonlinear behavior of these structures. The von-Mises stress contour extracted from FEM shows that the critical area of these soft structures is at the interface of the layers which experiences the maximum stress, and this area is most likely to rupture in large deformations.Mimetic finite differences for boundaries misaligned with grid nodeshttps://zbmath.org/1516.650652023-09-19T14:22:37.575876Z"Belousov, Danila"https://zbmath.org/authors/?q=ai:belousov.danila"Lisitsa, Vadim"https://zbmath.org/authors/?q=ai:lisitsa.vadimSummary: This paper considers the problem of mimetic staggered-grid finite difference scheme construction in the case of a boundary misaligned with the grid points. We show that depending on the reciprocal positions of the nearest grid point and whether the point is inside or outside the domain, two cases should be considered separately. In both cases, we present general rules to construct the mimetic difference operators to approximate the gradient and divergence operators. In addition, we investigate the stability criterion of the explicit in time scheme, which employs derived operators to approximate spatial derivatives. We show that, depending on the reciprocal positions of the nearest grid point, the Courant number may be smaller by 7\% than that of the initial value problem approximation.Robust numerical methods for nonlocal (and local) equations of porous medium type. II: Schemes and experimentshttps://zbmath.org/1516.650662023-09-19T14:22:37.575876Z"del Teso, Félix"https://zbmath.org/authors/?q=ai:del-teso.felix"Endal, Jørgen"https://zbmath.org/authors/?q=ai:endal.jorgen"Jakobsen, Espen R."https://zbmath.org/authors/?q=ai:jakobsen.espen-robstadSummary: We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations \(\partial_tu-\mathfrak L[\varphi(u)]=f(x,t)\) in \(\mathbb R^N\times(0,T)\), where \(\mathfrak L\) is a general symmetric Lévy-type diffusion operator. Included are both local and nonlocal problems with, e.g., \(\mathfrak L=\Delta\) or \(\mathfrak L=-(-\Delta)^{\frac{\alpha}{2}}\), \(\alpha\in(0,2)\), and porous medium, fast diffusion, and Stefan-type nonlinearities \(\phi\). By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are \(L^p\)-stable for \(p\in[1,\infty]\), compact, and convergent in \(C([0,T];L_{\mathrm{loc}}^p(\mathbb R^N))\) for \(p\in[1,\infty)\). The first part of this project is given in [\url{arXiv:1801.07148}] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of Part I [the authors, SIAM J. Numer. Anal. 57, No. 5, 2266--2299 (2019; Zbl 1428.65005)] apply and testing the schemes numerically. Our examples include fractional diffusions of different orders and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems.A parameter uniform essentially first-order convergent numerical method for a parabolic system of singularly perturbed differential equations of reaction-diffusion type with initial and Robin boundary conditionshttps://zbmath.org/1516.650672023-09-19T14:22:37.575876Z"Ishwariya, R."https://zbmath.org/authors/?q=ai:ishwariya.raj"Miller, J. J. H."https://zbmath.org/authors/?q=ai:miller.john-j-h"Valarmathi, S."https://zbmath.org/authors/?q=ai:valarmathi.sigamaniSummary: In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The components of the solution \(\vec{u}\) of this system are smooth, whereas the components of \(\frac{\partial \vec{u}}{\partial x}\) exhibit parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.Construction of the high order accurate generalized finite difference schemes for inviscid compressible flowshttps://zbmath.org/1516.650682023-09-19T14:22:37.575876Z"Li, Xue-Li"https://zbmath.org/authors/?q=ai:li.xueli"Ren, Yu-Xin"https://zbmath.org/authors/?q=ai:ren.yuxin"Li, Wanai"https://zbmath.org/authors/?q=ai:li.wanaiSummary: The construction of high order accurate generalized finite difference method for inviscid compressible flows still remains an open problem in the literature. In this paper, the high order accurate generalized finite difference schemes have been developed based on the high order reconstruction and the high order numerical flux evaluation on a local cloud of points. The WBAP limiter based on the secondary reconstruction is used to suppress oscillations near discontinuities. The implementation of high order accurate boundary conditions is of critical importance in the construction of high order schemes. A new method is proposed for the high order accurate boundary treatment. Several standard test cases are solved to validate the accuracy, efficiency and shock capturing capability of the proposed high order schemes.Convergence rates of the front tracking method for conservation laws in the Wasserstein distanceshttps://zbmath.org/1516.650692023-09-19T14:22:37.575876Z"Solem, Susanne"https://zbmath.org/authors/?q=ai:solem.susanneSummary: We prove that front tracking approximations to scalar conservation laws with convex fluxes converge at a rate of \(\Delta x^2\) in the 1-Wasserstein distance \(W_1\). Assuming positive initial data, we also show that the approximations converge at a rate of \(\Delta x\) in the \(\infty\)-Wasserstein distance \(W_\infty\). Moreover, from a simple interpolation inequality between \(W_1\) and \(W_\infty\) we obtain convergence rates in all the \(p\)-Wasserstein distances: \(\Delta x^{1+1/p}\), \(p\in[1,\infty]\).Efficient numerical computation of time-fractional nonlinear Schrödinger equations in unbounded domainhttps://zbmath.org/1516.650702023-09-19T14:22:37.575876Z"Zhang, Jiwei"https://zbmath.org/authors/?q=ai:zhang.jiwei"Li, Dongfang"https://zbmath.org/authors/?q=ai:li.dongfang"Antoine, Xavier"https://zbmath.org/authors/?q=ai:antoine.xavierSummary: The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [\textit{J. Zhang} at al., Phys. Rev. E (3) 78, No. 2, Article ID 026709, 8 p. (2008; \url{doi:10.1103/PhysRevE.78.026709}); Phys. Rev. E (3) 79, No. 4, Article ID 046711, 8 p. (2009; \url{doi:10.1103/PhysRevE.79.046711})] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate \(\mathcal{O}(h^2+\tau)\) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equationshttps://zbmath.org/1516.650712023-09-19T14:22:37.575876Z"Zhao, Chengchao"https://zbmath.org/authors/?q=ai:zhao.chengchao"Liu, Nan"https://zbmath.org/authors/?q=ai:liu.nan"Ma, Yuheng"https://zbmath.org/authors/?q=ai:ma.yuheng"Zhang, Jiwei"https://zbmath.org/authors/?q=ai:zhang.jiweiSummary: In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. the ratio less than \(4.8645\), and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution obtained by BDF1 (\textit{i.e.}, backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results.The Cartesian grid active flux method with adaptive mesh refinementhttps://zbmath.org/1516.650722023-09-19T14:22:37.575876Z"Calhoun, Donna"https://zbmath.org/authors/?q=ai:calhoun.donna-a"Chudzik, Erik"https://zbmath.org/authors/?q=ai:chudzik.erik"Helzel, Christiane"https://zbmath.org/authors/?q=ai:helzel.christianeSummary: We present the first implementation of the Active Flux method on adaptively refined Cartesian grids. The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method has a compact stencil in space and time and good stability properties. The method is implemented as a new solver in ForestClaw, a software for parallel adaptive mesh refinement of patch-based solvers. On each Cartesian grid patch the single grid Active Flux method can be applied. The exchange of data between grid patches is organised via ghost cells. The local stencil in space and time and the availability of the point values that are used for the reconstruction, leads to an efficient implementation. The resulting method is third order accurate, conservative and allows the use of subcycling in time.An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniqueshttps://zbmath.org/1516.650732023-09-19T14:22:37.575876Z"Demo, Nicola"https://zbmath.org/authors/?q=ai:demo.nicola"Ortali, Giulio"https://zbmath.org/authors/?q=ai:ortali.giulio"Gustin, Gianluca"https://zbmath.org/authors/?q=ai:gustin.gianluca"Rozza, Gianluigi"https://zbmath.org/authors/?q=ai:rozza.gianluigi"Lavini, Gianpiero"https://zbmath.org/authors/?q=ai:lavini.gianpieroSummary: This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretization of a two-phase (water and air) fluid. Since the fluid dynamics model can result very expensive -- especially dealing with complex industrial geometries -- we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique. Thanks to the quick approximation, a genetic optimization algorithm becomes feasible to converge towards the optimal shape.Quadratic stability of flux limitershttps://zbmath.org/1516.650742023-09-19T14:22:37.575876Z"Després, Bruno"https://zbmath.org/authors/?q=ai:despres.brunoThe author proposes a novel approach to study the quadratic stability of 2D flux limiters for non expansive transport equations. The approach is developed for the constant coefficient case on a cartesian grid. The convergence of the fully discrete nonlinear scheme is established in 2D with a rate not less than \(O(\Delta x^{\frac{1}{2}})\) in quadratic norm. A new nonlinear scheme with corner correction is proposed. The scheme is formally second-order accurate away from characteristics points, satisfies the maximum principle, and is proved to be convergent in quadratic norm. Some numerical tests are presented to support the theoretical results.
Reviewer: Abdallah Bradji (Annaba)An adaptive finite volume method for the incompressible Navier-Stokes equations in complex geometrieshttps://zbmath.org/1516.650752023-09-19T14:22:37.575876Z"Trebotich, David"https://zbmath.org/authors/?q=ai:trebotich.david-p"Graves, Daniel T."https://zbmath.org/authors/?q=ai:graves.daniel-tThe authors present a conservative, second-order accurate finite volume method to solve the incompressible Navier-Stokes equations in complex geometries. The method is based on an embedded boundary approach that makes use of the discrete form of the divergence theorem to discretize the solution in irregular control volumes resulting from the intersection of solid boundaries with a regular, Cartesian grid. The method is a finite difference approach in regular cells away from the boundary. Thanks to the use of a volume-weighted scheme, the method avoids the small-cell problem associated with cut cell methods and a conservative cell-centered gradient for approximate projections. The embedded boundary method is coupled with AMR (Adaptive Mesh Refinement) to provide a high-performance, high-resolution simulation tool for modeling multiscale, multiphysics problems in complex geometries. The Helmholtz equations resulting from viscous source terms are advanced in time by the Crank-Nicolson method, which reduces solver runtime compared to other second-order time integrators by a half. Incompressibility is enforced by a second-order approximate projection method that makes use of a new conservative cell-centered gradient in cut cells that is consistent with the volume-weighted scheme. The robustness of the algorithm is demonstrated for a wide range of Reynolds numbers and flow geometries, from creeping flow in realistic pore space to transitional flows past bluff bodies to turbulent pipe flow.
Reviewer: Abdallah Bradji (Annaba)Kinetic energy-free Hartree-Fock equations: an integral formulationhttps://zbmath.org/1516.650762023-09-19T14:22:37.575876Z"Jensen, Stig Rune"https://zbmath.org/authors/?q=ai:jensen.stig-rune"Durdek, Antoine"https://zbmath.org/authors/?q=ai:durdek.antoine"Bjørgve, Magnar"https://zbmath.org/authors/?q=ai:bjorgve.magnar"Wind, Peter"https://zbmath.org/authors/?q=ai:wind.peter"Flå, Tor"https://zbmath.org/authors/?q=ai:fla.tor"Frediani, Luca"https://zbmath.org/authors/?q=ai:frediani.lucaThe authors of this work designed and implemented a solver for Hartree-Fock (HF) equations which are considered as cornerstones of quantum chemistry. The equations are reformulated as coupled integral equations therefore making it possible to find the solution without implementing a differential operator. Convolution and derivative operators related to Multiwavelet (MW) framework were discussed and the descriptions of the Fock operator and iterative solution of the minimization problem are given. Usual strategies for obtaining the minimizer involves a finite basis representation of blocks of the Fock matrix. An integral representation of the HF equations is considered due to the concerns involving the use of differential operators within the MW approach. A detailed description of the integral representation that doesn't require the explicit use of the kinetic energy operator and gives rise to development of efficient iterative algorithms is given. Extension of the procedure for a many-electron system is also discussed. Calculation of Fock matrix and energy, orbital orthonormalization, as well as the implementation details and the algorithm are provided. Numerical results with hydrogen and beryllium atoms demonstrating robust convergence patterns are presented.
Reviewer: Baasansuren Jadamba (Rochester)Effect of periodic heat transfer on the transient thermal behavior of a convective-radiative fully wet porous moving trapezoidal finhttps://zbmath.org/1516.650772023-09-19T14:22:37.575876Z"Gireesha, B. J."https://zbmath.org/authors/?q=ai:gireesha.bijjanal-jayanna"Keerthi, M. L."https://zbmath.org/authors/?q=ai:keerthi.m-lSummary: A moving trapezoidal profiled convective-radiative porous longitudinal fin wetted in a single-phase fluid is considered in the current article. The periodic variation in the fin base temperature is taken into account along with the temperature sensitive thermal conductivity and convective heat transfer coefficients. The modeled problem, which is resolved into a non-linear partial differential equation (PDE), is made dimensionless and solved by employing the finite difference method (FDM). The results are displayed through graphs and discussed. The effects of amplitude, frequency of oscillation, wet nature, Peclet number, and other relevant quantities on the distribution of temperature through the fin length and with the dimensionless time are investigated. It is deciphered that the periodic heat transfer gives rise to the wavy nature of the fin thermal profile against time. The analysis is beneficial in the design of fin structures for applications like solar collectors, space/airborne applications, and refrigeration industries.Inverse spectral problem for a damped wave operatorhttps://zbmath.org/1516.650782023-09-19T14:22:37.575876Z"Bao, Gang"https://zbmath.org/authors/?q=ai:bao.gang"Xu, Xiang"https://zbmath.org/authors/?q=ai:xu.xiang"Zhai, Jian"https://zbmath.org/authors/?q=ai:zhai.jianSummary: This paper proposes a new and efficient numerical algorithm for recovering the damping coefficient from the spectrum of a damped wave operator. The algorithm is based on inverting a sequence of \textit{trace formulas}, which are deduced by a recursive formula, bridging geometrical and spectrum information explicitly in terms of Fredholm integral equations. Numerical examples are presented to illustrate the efficiency of the proposed algorithm.Weak-norm posterior contraction rate of the 4DVAR method for linear severely ill-posed problemshttps://zbmath.org/1516.650792023-09-19T14:22:37.575876Z"Ding, Litao"https://zbmath.org/authors/?q=ai:ding.litao"Lu, Shuai"https://zbmath.org/authors/?q=ai:lu.shuai"Cheng, Jin"https://zbmath.org/authors/?q=ai:cheng.jinSummary: Inspired by the artificial dynamic, we solve the linear severely ill-posed problems based on the 4DVAR method arising in data assimilation. To obtain the consistency of the posterior distribution in the weak norm by a fractional order of the forward operator, we are interested in asymptotic behavior of the weak norm squared posterior contraction (\(\operatorname{SPC}_L\)) function. By assuming exponentially decaying spectrum of the forward operator, the estimate of the \(\operatorname{SPC}_L\) function is established for a Sobolev-like source condition. Such a result is further extended to a general source condition. In both cases, we verify that the severe ill-posedness of the inverse problem can be reduced to moderate ill-posedness by the weak norm.Symmetric interior penalty discontinuous Galerkin discretizations and block preconditioning for heterogeneous Stokes flowhttps://zbmath.org/1516.650802023-09-19T14:22:37.575876Z"Charrier, D. E."https://zbmath.org/authors/?q=ai:charrier.dominic-etienne"May, D. A."https://zbmath.org/authors/?q=ai:may.dave-a|may.d-an"Schnepp, S. M."https://zbmath.org/authors/?q=ai:schnepp.sascha-mSummary: Provable stable arbitrary order symmetric interior penalty (SIP) discontinuous Galerkin discretizations of heterogeneous, incompressible Stokes flow utilizing \(Q^2_k\)-\(Q_{k-1}\) elements and hierarchical Legendre basis polynomials are developed and investigated. For solving the resulting linear system, a block preconditioned iterative method is proposed. The nested viscous problem is solved by a \(hp\)-multilevel preconditioned Krylov subspace method. For the \(p\)-coarsening, a two-level method utilizing element-block Jacobi preconditioned iterations as a smoother is employed. Piecewise bilinear (\(Q^2_1\)) and piecewise constant (\(Q^2_0\)) \(p\)-coarse spaces are considered. Finally, Galerkin \(h\)-coarsening is proposed and investigated for the two \(p\)-coarse spaces considered. Through a number of numerical experiments, we demonstrate that utilizing the \(Q^2_1\) coarse space results in the most robust \(hp\)-multigrid method for heterogeneous Stokes flow. Using this \(Q^2_1\) coarse space we observe that the convergence of the overall Stokes solver appears to be robust with respect to the jump in the viscosity and only mildly depending on the polynomial order \(k\). It is demonstrated and supported by theoretical results that the convergence of the SIP discretizations and the iterative methods rely on a sharp choice of the penalty parameter based on local values of the viscosity.Non-overlapping Schwarz waveform-relaxation for nonlinear advection-diffusion equationshttps://zbmath.org/1516.650812023-09-19T14:22:37.575876Z"Gander, Martin J."https://zbmath.org/authors/?q=ai:gander.martin-j"Lunowa, Stephan B."https://zbmath.org/authors/?q=ai:lunowa.stephan-benjamin"Rohde, Christian"https://zbmath.org/authors/?q=ai:rohde.christianSummary: Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.Iterative methods with nonconforming time grids for nonlinear flow problems in porous mediahttps://zbmath.org/1516.650822023-09-19T14:22:37.575876Z"Thi-Thao-Phuong Hoang"https://zbmath.org/authors/?q=ai:thi-thao-phuong-hoang."Pop, Iuliu Sorin"https://zbmath.org/authors/?q=ai:pop.iuliu-sorinSummary: Partially saturated flow in a porous medium is typically modeled by the Richards equation, which is nonlinear, parabolic and possibly degenerated. This paper presents domain decomposition-based numerical schemes for the Richards equation, in which different time steps can be used in different subdomains. Two global-in-time domain decomposition methods are derived in mixed formulations: the first method is based on the physical transmission conditions and the second method is based on equivalent Robin transmission conditions. For each method, we use substructuring techniques to rewrite the original problem as a nonlinear problem defined on the space-time interfaces between the subdomains. Such a space-time interface problem is linearized using Newton's method and then solved iteratively by GMRES; each GMRES iteration involves parallel solution of time-dependent problems in the subdomains. Numerical experiments in two dimensions are carried out to verify and compare the convergence and accuracy of the proposed methods with local time stepping.Uncoupling evolutionary groundwater-surface water flows: stabilized mixed methods in both porous media and fluid regionshttps://zbmath.org/1516.650832023-09-19T14:22:37.575876Z"Al Mahbub, Md. Abdullah"https://zbmath.org/authors/?q=ai:al-mahbub.md-abdullah"Shan, Li"https://zbmath.org/authors/?q=ai:shan.li"Zheng, Haibiao"https://zbmath.org/authors/?q=ai:zheng.haibiaoSummary: This paper considers the robust numerical methods for solving the time-dependent Stokes-Darcy multiphysics problem that can be implemented by use of existing surface water and groundwater codes. Porous media problem for the groundwater flow is preferable to employ the mixed discretization due to their superior conservation property and the convenience to compute flux on the large domain with relatively coarse meshes. However, the theory of mixed spatial discretizations for the time-dependent problems is far less developed than the non-mixed approaches, even for the one domain problems. Herein, we develop a stabilized mixed discretization technique for the porous media problem coupled with the fluid region across an interface with the physically appropriate coupling conditions. Time discretization is constructed to allow a non-iterative splitting of the coupled problem into two subproblems. The stability and convergence analysis of the coupled and decoupled algorithms are derived rigorously. If the time scale is bounded by a constant which only depends on the physical parameters, we prove the unconditional stability of both schemes. Four numerical experiments are conducted to show the efficiency and accuracy of the numerical methods, which illustrate the exclusive features of the Stokes-Darcy interface system.Some observations on the interaction between linear and nonlinear stabilization for continuous finite element methods applied to hyperbolic conservation lawshttps://zbmath.org/1516.650842023-09-19T14:22:37.575876Z"Burman, Erik"https://zbmath.org/authors/?q=ai:burman.erikThe author investigates the interaction between linear and nonlinear stabilizations of continuous finite element methods for hyperbolic conservation laws. As a result, he gives the error estimate of order \(O(h^{k+1/2})\) (\(h\) is a mesh size) between the approximate solution and the sufficient smooth solution belonging to \(L^{\infty}(0,T;H^{k+1}(\Omega)) \cap H^1(0,T;H^{k}(\Omega)) \cap L^{\infty}(0,T;W^{1,\infty}(\Omega))\) for the scalar linear hyperbolic transport equation. In the numerical scheme, two stabilizers are used. One comes from the gradient jump penalty (GJP) method and the other comes from the nonlinear artificial viscosity. Although the original equation is linear, the result is not trivial because the scheme is nonlinear due to the nonlinear stabilizer. The result is obtained by careful and interesting observations of the interaction of linear and nonlinear stabilizers.
Reviewer: Shuji Yoshikawa (Oita)Fully discrete heterogeneous multiscale method for parabolic problems with multiple spatial and temporal scaleshttps://zbmath.org/1516.650852023-09-19T14:22:37.575876Z"Eckhardt, Daniel"https://zbmath.org/authors/?q=ai:eckhardt.daniel-q"Verfürth, Barbara"https://zbmath.org/authors/?q=ai:verfurth.barbaraProblems with macroscopic and microscopic spatial and temporal scales occur in various phenomena and materials. They are challenging from a numerical point of view. In this work, the Heterogeneous Multiscale Method (HMM) is used for numerical homogenization of a parabolic problem with several temporal and spatial scales. A suitable discretization of the cell problems is proposed and rigorous error estimates are derived for the resulting fully discrete HMM, whereas the order of the mesh size and the time step on the one hand and the period on the other hand are balanced. The convergence rate with respect to the time step and mesh width is investigated for the macrodiscretization as well as for the microdiscretization in numerical experiments.
Reviewer: Bülent Karasözen (Ankara)POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solutionhttps://zbmath.org/1516.650862023-09-19T14:22:37.575876Z"García-Archilla, Bosco"https://zbmath.org/authors/?q=ai:garcia-archilla.bosco"John, Volker"https://zbmath.org/authors/?q=ai:john.volker"Novo, Julia"https://zbmath.org/authors/?q=ai:novo.juliaSummary: In this paper we study the influence of including snapshots that approach the velocity time derivative in the numerical approximation of the incompressible Navier-Stokes equations by means of proper orthogonal decomposition (POD) methods. Our set of snapshots includes the velocity approximation at the initial time from a full order mixed finite element method (FOM) together with approximations to the time derivative at different times. The approximation at the initial velocity can be replaced by the mean value of the velocities at the different times so that when implementing the method to the fluctuations, as done mostly in practice, only approximations to the time derivatives are included in the set of snapshots. For the POD method we study the differences between projecting onto \(L^2\) and \(H^1\). In both cases pointwise in time error bounds can be proved. Including grad-div stabilization in both the FOM and the POD methods, error bounds with constants independent of inverse powers of the viscosity can be obtained.A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEshttps://zbmath.org/1516.650872023-09-19T14:22:37.575876Z"Haasdonk, Bernard"https://zbmath.org/authors/?q=ai:haasdonk.bernard"Kleikamp, Hendrik"https://zbmath.org/authors/?q=ai:kleikamp.hendrik"Ohlberger, Mario"https://zbmath.org/authors/?q=ai:ohlberger.mario"Schindler, Felix"https://zbmath.org/authors/?q=ai:schindler.felix"Wenzel, Tizian"https://zbmath.org/authors/?q=ai:wenzel.tizianSummary: We present a new surrogate modeling technique for efficient approximation of input-output maps governed by parametrized PDEs. The model is hierarchical as it is built on a full order model, reduced order model (ROM), and machine learning (ML) model chain. The model is adaptive in the sense that the ROM and ML model are adapted on the fly during a sequence of parametric requests to the model. To allow for a certification of the model hierarchy, as well as to control the adaptation process, we employ rigorous a posteriori error estimates for the ROM and ML models. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. We demonstrate the efficiency of the modeling chain on a Monte Carlo and a parameter-optimization example. Here, the ROM is instantiated by Reduced Basis methods, and the ML model is given by a neural network or by a kernel model using vectorial kernel orthogonal greedy algorithms.Discontinuous Galerkin methods with generalized numerical fluxes for the Vlasov-viscous Burgers' systemhttps://zbmath.org/1516.650882023-09-19T14:22:37.575876Z"Hutridurga, Harsha"https://zbmath.org/authors/?q=ai:hutridurga.harsha"Kumar, Krishan"https://zbmath.org/authors/?q=ai:kumar.krishan"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumarSummary: In this paper, semi-discrete numerical scheme for the approximation of the periodic Vlasov-viscous Burgers' system is developed and analyzed. The scheme is based on the coupling of discontinuous Galerkin approximations for the Vlasov equation and local discontinuous Galerkin approximations for the viscous Burgers' equation. Both these methods use generalized numerical fluxes. The proposed scheme is both mass and momentum conservative. Based on generalized Gauss-Radau projections, optimal rates of convergence in the case of smooth compactly supported initial data are derived. Finally, computational results confirm our theoretical findings.Positivity-preserving Lax-Wendroff discontinuous Galerkin schemes for quadrature-based moment-closure approximations of kinetic modelshttps://zbmath.org/1516.650892023-09-19T14:22:37.575876Z"Johnson, Erica R."https://zbmath.org/authors/?q=ai:johnson.erica-r"Rossmanith, James A."https://zbmath.org/authors/?q=ai:rossmanith.james-a"Vaughan, Christine"https://zbmath.org/authors/?q=ai:vaughan.christineSummary: The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor-corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear algebraic system in this prediction is solved using a Picard iteration. The correction is a straightforward explicit update based on the time-integral of the evolution equation, where the space-time prediction replaces all instances of the exact solution. In the absence of limiters, the high-order scheme does not guarantee that solutions remain in the convex set over which HyQMOM is moment-realizable. To overcome this, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set of realizable solutions, thus guaranteeing the hyperbolicity of the system. We develop positivity-preserving limiters in both the prediction and correction steps and an oscillation limiter that damps unphysical oscillations near shocks. We also develop a novel extension of this scheme to include a BGK collision operator; the proposed method is shown to be asymptotic-preserving in the high-collision limit. The HyQMOM and the HyQMOM-BGK solvers are verified on several test cases, demonstrating high-order accuracy on smooth problems and shock-capturing capability on problems with shocks. The asymptotic-preserving property of the HyQMOM-BGK solver is also numerically verified.Optimal a priori error estimate of relaxation-type linear finite element method for nonlinear Schrödinger equationhttps://zbmath.org/1516.650902023-09-19T14:22:37.575876Z"Liu, Huini"https://zbmath.org/authors/?q=ai:liu.huini"Yi, Nianyu"https://zbmath.org/authors/?q=ai:yi.nianyuSummary: In this paper, we study, analyze and numerically validate a conservative relaxation-type linear finite element method (FEM) for the nonlinear Schrödinger equation. The method avoids solving the complex nonlinear system and preserves the discrete mass and energy. A key to our analysis is that the errors are split into the temporal error and the spatial error by introducing the corresponding time-discrete system. Therefore, we derive the optimal \(L^2\) and semi-\(H^1\) error estimates without any coupling condition between time step \(\tau\) and space size \(h\). Some numerical experiments not only demonstrate the method's optimal convergence rates but also confirm the conservation laws during long time simulations.Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw systemhttps://zbmath.org/1516.650912023-09-19T14:22:37.575876Z"Liu, Yuan"https://zbmath.org/authors/?q=ai:liu.yuan"Chen, Wenbin"https://zbmath.org/authors/?q=ai:chen.wenbin"Wang, Cheng"https://zbmath.org/authors/?q=ai:wang.cheng.1"Wise, Steven M."https://zbmath.org/authors/?q=ai:wise.steven-mSummary: We present and analyze a mixed finite element numerical scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in [\textit{X. Feng} and \textit{S. Wise}, SIAM J. Numer. Anal. 50, No. 3, 1320--1343 (2012; Zbl 1426.76258)], with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard \(\ell ^\infty (0,T;L^2) \cap \ell ^2 (0,T; H^2)\) error estimate, we perform a discrete \(\ell ^\infty (0,T; H^1) \cap \ell ^2 (0,T; H^3 )\) error estimate for the phase variable, through an \(L^2\) inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step \(\tau \) in terms of the spatial resolution \(h\)) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo-Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian \(\Delta _h\) of the numerical solution, such that \(\Delta _h \phi \in S_h\), for every \(\phi \in S_h\), where \(S_h\) is the finite element space.Error analysis of a unconditionally stable BDF2 finite element scheme for the incompressible flows with variable densityhttps://zbmath.org/1516.650922023-09-19T14:22:37.575876Z"Li, Yuan"https://zbmath.org/authors/?q=ai:li.yuan.1"An, Rong"https://zbmath.org/authors/?q=ai:an.rongSummary: Based on an equivalent form of the variable density flows, we propose and study a second-order linearized finite element scheme for the approximation of the three-dimensional incompressible Navier-Stokes equations with variable density, where the two-step backward differentiation formula is used in the discretization of time derivative. It is shown that the proposed finite element scheme is unconditionally stable in the sense that the discrete energy inequalities hold without any condition on the time step size and mesh size. By a rigorous error analysis, the optimal second-order convergence rate is proved in \(L^2\)-norm. Finally, numerical results are provided to confirm our theoretical analysis.A mixed discontinuous Galerkin method for a linear viscoelasticity problem with strongly imposed symmetryhttps://zbmath.org/1516.650932023-09-19T14:22:37.575876Z"Meddahi, Salim"https://zbmath.org/authors/?q=ai:meddahi.salim"Ruiz-Baier, Ricardo"https://zbmath.org/authors/?q=ai:ruiz-baier.ricardoSummary: We propose and rigorously analyze a semi- and fully discrete discontinuous Galerkin method for an initial and boundary value problem describing inertial viscoelasticity in terms of elastic and viscoelastic stress components and with mixed boundary conditions. The arbitrary-order spatial discretization imposes strongly the symmetry of the stress tensor, and it is combined with a Newmark trapezoidal rule as a time-advancing scheme. We establish stability and convergence properties, and the theoretical findings are further confirmed via illustrative numerical simulations in 2 and 3 dimensions.Unisolvence of symmetric node patterns for polynomial spaces on the simplexhttps://zbmath.org/1516.650942023-09-19T14:22:37.575876Z"Mulder, W. A."https://zbmath.org/authors/?q=ai:mulder.william-a|mulder.wim-aSummary: Finite elements with polynomial basis functions on the simplex with a symmetric distribution of nodes should have a unique polynomial representation. Unisolvence not only requires that the number of nodes equals the number of independent polynomials spanning a polynomial space of a given degree, but also that the Vandermonde matrix controlling their mapping to the Lagrange interpolating polynomials can be inverted. Here, a necessary condition for unisolvence is presented for polynomial spaces that have non-decreasing degrees when going from the edges and the various faces to the interior of the simplex. It leads to a proof of a conjecture on a necessary condition for unisolvence, requiring the node pattern to be the same as that of the regular simplex.A kinetic energy preserving DG scheme based on Gauss-Legendre pointshttps://zbmath.org/1516.650952023-09-19T14:22:37.575876Z"Ortleb, Sigrun"https://zbmath.org/authors/?q=ai:ortleb.sigrunSummary: In the context of numerical methods for conservation laws, not only the preservation of the primary conserved quantities can be of interest, but also the balance of secondary ones such kinetic energy in case of the Euler equations of gas dynamics. In this work, we construct a kinetic energy preserving discontinuous Galerkin method on Gauss-Legendre nodes based on the framework of summation-by-parts operators. For a Gauss-Legendre point distribution, boundary terms require special attention. In fact, stability problems will be demonstrated for a combination of skew-symmetric and boundary terms that disagrees with exclusively interior nodal sets. We will theoretically investigate the required form of the corresponding boundary correction terms in the skew-symmetric formulation leading to a conservative and consistent scheme. In numerical experiments, we study the order of convergence for smooth solutions, the kinetic energy balance and the behaviour of different variants of the scheme applied to
an acoustic pressure wave and a viscous shock tube. Using Gauss-Legendre nodes results in a more accurate approximation in our numerical experiments for viscous compressible flow. Moreover, for two-dimensional decaying homogeneous turbulence, kinetic energy preservation yields a better representation of the energy spectrum.A structure-preserving parametric finite element method for area-conserved generalized curvature flowhttps://zbmath.org/1516.650962023-09-19T14:22:37.575876Z"Pei, Lifang"https://zbmath.org/authors/?q=ai:pei.lifang"Li, Yifei"https://zbmath.org/authors/?q=ai:li.yifeiSummary: A structure-preserving numerical method is presented and analyzed to simulate the motion of closed curves governed by area-conserved generalized curvature flow. We first propose a variational formulation and rigorously prove that it preserves two geometric structures of enclosed area conservation and perimeter decrease. Then the parametric finite element method is adopted to develop a semi-discrete scheme. Both the area conservation and perimeter decrease structure properties are strictly proven for the semi-discrete scheme by utilizing the Cauchy-Schwarz inequality and power mean inequality. On this basis, a fully discrete scheme is established by further taking the backward Euler method in the temporal direction and discretizing the unit normal vector appropriately. We give a rigorous proof that the scheme is structure-preserving at the discretized level. Finally, numerical results verify the structure-preserving property and reveal that the proposed method has second-order accuracy and enjoys asymptotic equal mesh distribution during the evolution.Exponential convergence of \(hp\)-time-stepping in space-time discretizations of parabolic PDEshttps://zbmath.org/1516.650972023-09-19T14:22:37.575876Z"Perugia, Ilaria"https://zbmath.org/authors/?q=ai:perugia.ilaria"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christoph"Zank, Marco"https://zbmath.org/authors/?q=ai:zank.marcoThis paper focus on solution of linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator written in divergence form. The time-analyticity of solutions is proven for time-analytic forcing term in polygonal/polyhedral domains. The discretization of the variational formulation of the problem is analyzed using an exponentially convergent time-discretization of \(hp\)-type, combined with a first-order Lagrangian FEM in the spatial domain, with corner-mesh refinement to account for the presence of spatial singularities. It is shown that this method behaves up to logarithmic terms to what standard FEM provide for one elliptic boundary value problem. Numerical examples are shown in for 1D case and for two-dimensional test case (L-shaped domain).
Reviewer: Petr Sváček (Praha)A fully-decoupled artificial compressible Crank-Nicolson-leapfrog time stepping scheme for the phase field model of two-phase incompressible flowshttps://zbmath.org/1516.650982023-09-19T14:22:37.575876Z"Qian, Lingzhi"https://zbmath.org/authors/?q=ai:qian.lingzhi"Wu, Chunya"https://zbmath.org/authors/?q=ai:wu.chunya"Cai, Huiping"https://zbmath.org/authors/?q=ai:cai.huiping"Feng, Xinlong"https://zbmath.org/authors/?q=ai:feng.xinlong"Qiao, Yuanyang"https://zbmath.org/authors/?q=ai:qiao.yuanyangSummary: In this paper, we consider efficient numerical approximations for the phase field model of two-phase incompressible flows. To develop easy-to-implement time stepping scheme, we introduce two types of nonlocal auxiliary variables to achieve highly efficient and fully-decoupled scheme based on the Crank-Nicolson-Leapfrog (CNLF) formula and artificial compression method. We prove that the scheme is linear and unconditionally energy stable. Ample numerical experiments are performed to demonstrate the accuracy, stability and efficiency.Error analysis of fully discrete scheme for the Cahn-Hilliard-magneto-hydrodynamics problemhttps://zbmath.org/1516.650992023-09-19T14:22:37.575876Z"Qiu, Hailong"https://zbmath.org/authors/?q=ai:qiu.hailongSummary: In this paper we analyze a fully discrete scheme for a general Cahn-Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stable and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence.Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertaintyhttps://zbmath.org/1516.651002023-09-19T14:22:37.575876Z"Sousedík, Bedřich"https://zbmath.org/authors/?q=ai:sousedik.bedrich"Lee, Kookjin"https://zbmath.org/authors/?q=ai:lee.kookjinThe authors of this work presented a method for linear stability analysis of systems with a stochastic parameter where the main focus is the Navier-Stokes equation with stochastic viscosity under stochastic Galerkin framework. The stability analysis leads to the solution of a stochastic eigenvalue problem, and the goal is to characterize the rightmost eigenvalue. First two sections of the paper include an introduction to the stochastic eigenvalue problem arising from the linear stability analysis and descriptions of sampling methods such as Monte-Carlo and stochastic collocation. The stochastic Galerkin method, Newton iterations, and a line-search modification aimed at improving the global convergence of the the Newton iteration are presented in the next section. Detailed descriptions of the algorithms such as mean-based, constraint mean-based, and constraint hierarchical Gauss-Seidel preconditioners are discussed. Application of the algorithms for the stability analysis of the Navier-Stokes equation where the stochastic viscosity parameter is assumed to have a stochastic expansion involving a set of deterministic spatial functions is discussed next. The final section included results of the numerical experiments with two model flow problems: one involving a flow around an obstacle and another with an expansion flow around a symmetric step. For the first problem, the rightmost eigenvalue is given by a complex conjugate pair and for the second problem it is real.
Reviewer: Baasansuren Jadamba (Rochester)Error estimates of a sphere-constraint-preserving numerical scheme for Ericksen-Leslie system with variable densityhttps://zbmath.org/1516.651012023-09-19T14:22:37.575876Z"Wang, Danxia"https://zbmath.org/authors/?q=ai:wang.danxia"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang|liu.fang.1|liu.fang.2|liu.fang.6"Jia, Hongen"https://zbmath.org/authors/?q=ai:jia.hongen"Zhang, Jianwen"https://zbmath.org/authors/?q=ai:zhang.jianwenSummary: In this paper, a new first-order, sphere-constraint-preserving numerical scheme is constructed for the Ericksen-Leslie equations with variable density. Firstly, by denoting the orientation field vector \(\boldsymbol{d}\) in the polar coordinate, we rewrite the Ericksen-Leslie system into an equivalent system such that the sphere constraint \(|\boldsymbol{d}| = 1\) can be still preserved at the discrete level. Secondly, we propose a first-order discretization scheme in time for the equivalent new system in which the numerical velocity and modified pressure are determined by a generalized Stokes problem at each time step. Then, the unconditional energy stability and the rigorous error estimates in time are both derived. Finally, some numerical simulations in two dimensions are provided to demonstrate the stability of energy and accuracy of the presented scheme.Provably positive central discontinuous Galerkin schemes via geometric quasilinearization for ideal MHD equationshttps://zbmath.org/1516.651022023-09-19T14:22:37.575876Z"Wu, Kailiang"https://zbmath.org/authors/?q=ai:wu.kailiang"Jiang, Haili"https://zbmath.org/authors/?q=ai:jiang.haili"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wangThe authors show the positivity of approximate solutions to the central discontinuous Galerkin (CDG) schemes for the compressible ideal magnetohydrodynamics (MHD) system. The solution for MHD should satisfy the positiveness of the fluid density, the thermal pressure, and the specific internal energy, from the physical point of view. On the other hand, it is not obvious that the properties are inherited in numerical schemes. Indeed, these are not satisfied in most numerical schemes of MHD. In this article, the authors reveal what condition is required for the positive property, by the geometric quasilinearization technique which transfers the nonlinear constraint into linear ones. As a result, the relation to the divergence-free (DF) condition is pointed out. Although the DF condition is naturally satisfied in the one-dimensional case, it is not trivial in the multi-dimensional case. To treat the issue, the authors also propose a new locally DF CDG scheme satisfying the above positive property.
Reviewer: Shuji Yoshikawa (Oita)Reformulated weak formulation and efficient fully discrete finite element method for a two-phase ferrohydrodynamics Shliomis modelhttps://zbmath.org/1516.651032023-09-19T14:22:37.575876Z"Zhang, Guo-dong"https://zbmath.org/authors/?q=ai:zhang.guodong.1"He, Xiaoming"https://zbmath.org/authors/?q=ai:he.xiaoming.1"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: The two-phase ferrohydrodynamics model consisting of Cahn-Hilliard equations, Navier-Stokes equations, magnetization equations, and magnetostatic equations is a highly nonlinear, coupled, and saddle point structural multiphysics PDE system. While various works exist to develop fully decoupled, linear, second-order in time, and unconditionally energy stable methods for simpler gradient flow models, existing ideas may not be applicable to this complex model or may be only applicable to part of this model. Therefore, significant challenges remain in developing corresponding efficient fully discrete numerical algorithms with the four above-mentioned desired properties, which will be addressed in this paper by dynamically incorporating several key ideas, including a reformulated weak formulation with special test functions for overcoming two major difficulties caused by the magnetostatic equation, the decoupling technique based on the ``zero-energy-contribution'' property to handle the coupled nonlinear terms, the second-order projection method for the Navier-Stokes equations, and the invariant energy quadratization (IEQ) method for the time marching. Among all these ideas, the reformulated weak formulation serves as a key bridge between the existing techniques and the challenges of the target model, with all of the four desired properties kept in mind. We demonstrate the well-posedness of the proposed scheme and rigorously show that the scheme is unconditionally energy stable. Extensive numerical simulations, including accuracy/stability tests, and several 2D/3D benchmark Rosensweig instability problems for ``spiking'' phenomena of ferrofluids are performed to verify the effectiveness of the scheme.Spectral discretization of an unsteady flow through a porous solidhttps://zbmath.org/1516.651042023-09-19T14:22:37.575876Z"Bernardi, Christine"https://zbmath.org/authors/?q=ai:bernardi.christine"Maarouf, Sarra"https://zbmath.org/authors/?q=ai:maarouf.sarra"Yakoubi, Driss"https://zbmath.org/authors/?q=ai:yakoubi.drissSummary: We consider the non stationary flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with mixed boundary conditions. Since the boundary pressure can present high variations, the permeability of the medium also depends on the pressure, so that the problem is nonlinear. We propose a discretization of this equation that combines Euler's implicit scheme in time and spectral methods in space. We prove optimal a priori error estimates and present some numerical experiments which confirm the interest of the discretization.Numerical methods for some nonlinear Schrödinger equations in soliton managementhttps://zbmath.org/1516.651052023-09-19T14:22:37.575876Z"He, Ying"https://zbmath.org/authors/?q=ai:he.ying"Zhao, Xiaofei"https://zbmath.org/authors/?q=ai:zhao.xiaofei.1|zhao.xiaofeiSummary: In this work, we consider the numerical solutions of a dispersion-managed nonlinear Schrödinger equation (DM-NLS) and a nonlinearity-managed NLS equation (NM-NLS). The two equations arise from the soliton managements in optics and matter waves, and they involve temporal discontinuous coefficients with possible frequent jumps and stiffness which cause numerical difficulties. We analyze to see the order reduction problems of some popular traditional methods, and then we propose a class of exponential-type dispersion-map integrators for both DM-NLS and NM-NLS. The proposed methods are explicit, efficient under Fourier pseudospectral discretizations and second order accurate in time regardless the jumps/jump-period in the dispersion map. The extension to the fast \& strong management regime of DM-NLS is made with uniform accuracy.A second-order low-regularity correction of Lie splitting for the semilinear Klein-Gordon equationhttps://zbmath.org/1516.651062023-09-19T14:22:37.575876Z"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyang"Schratz, Katharina"https://zbmath.org/authors/?q=ai:schratz.katharina"Zivcovich, Franco"https://zbmath.org/authors/?q=ai:zivcovich.francoSummary: The numerical approximation of nonsmooth solutions of the semilinear Klein-Gordon equation in the \(d\)-dimensional space, with \(d = 1\), \(2\), \(3\), is studied based on the discovery of a new cancellation structure in the equation. This cancellation structure allows us to construct a low-regularity correction of the Lie splitting method (\textit{i.e.}, exponential Euler method), which can significantly improve the accuracy of the numerical solutions under low-regularity conditions compared with other second-order methods. In particular, the proposed time-stepping method can have second-order convergence in the energy space under the regularity condition \((u, \partial_t u) \in L^\infty (0,T; H^{1 + \frac{d}{4}} \times H^{\frac{d}{4}})\). In one dimension, the proposed method is shown to have almost \(\frac{4}{3}\)-order convergence in \(L^\infty (0, T; H^1 \times L^2)\) for solutions in the same space, \textit{i.e.}, no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. The numerical experiments show that the proposed time-stepping method is much more accurate than previously proposed methods for approximating the time dynamics of nonsmooth solutions of the semilinear Klein-Gordon equation.Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysicshttps://zbmath.org/1516.651072023-09-19T14:22:37.575876Z"Ling, Dan"https://zbmath.org/authors/?q=ai:ling.dan"Mao, Zhiping"https://zbmath.org/authors/?q=ai:mao.zhipingSummary: The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.An asymptotic preserving scheme for Lévy-Fokker-Planck equation with fractional diffusion limithttps://zbmath.org/1516.651082023-09-19T14:22:37.575876Z"Xu, Wuzhe"https://zbmath.org/authors/?q=ai:xu.wuzhe"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.58|wang.li.6Summary: In this paper, we develop a numerical method for the Lévy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold nonlocality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other arises from long-time/small mean-free-path scaling, which introduces stiffness into the equation. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply the Chebyshev polynomial based pseudo-spectral method. To treat the multiple scales, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically. Finally, the efficiency and accuracy of our scheme are illustrated by a suite of numerical examples.Stochastic Galerkin methods for time-dependent radiative transfer equations with uncertain coefficientshttps://zbmath.org/1516.651092023-09-19T14:22:37.575876Z"Zheng, Chuqi"https://zbmath.org/authors/?q=ai:zheng.chuqi"Qiu, Jiayu"https://zbmath.org/authors/?q=ai:qiu.jiayu"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin"Zhong, Xinghui"https://zbmath.org/authors/?q=ai:zhong.xinghuiSummary: The generalized polynomial chaos (gPC) method is one of the most popular method for uncertainty quantification. Being essentially a spectral approach, the gPC method exhibits the spectral convergence rate which heavily depends on the regularity of the solution in the random space. Many regularity studies have been made for stochastic elliptic and parabolic equations while regularities studies of stochastic hyperbolic equations has long been infeasible due to its intrinsic difficulties. In this paper, we investigate the impact of uncertainty on the time-dependent radiative transfer equation (RTE) with nonhomogeneous boundary conditions, which sits somewhere between hyperbolic and parabolic equations. We theoretically prove the a-priori bound of the solution, its continuity with respect to the scattering coefficient, and its regularity in the random space. These studies can serve as a building block in understanding the influence of uncertainties in the passage from hyperbolic to parabolic equations. Moreover, we vigorously justify the validity of the gPC expansion ansatz based on the regularity study. Then the stochastic Galerkin method of the gPC approach is employed to discretize the random variable. We further conduct a delicate analysis to show the exponential decay rate of the gPC coefficients and establish the error estimates of the stochastic Galerkin approximation for both one-dimensional and multi-dimensional random space cases. Numerical tests are presented to verify our analytical results.Bayesian calibration of a numerical code for prediction. Theory of code calibration and application to the prediction of a photovoltaic power plant electricity productionhttps://zbmath.org/1516.651102023-09-19T14:22:37.575876Z"Carmassi, Mathieu"https://zbmath.org/authors/?q=ai:carmassi.mathieu"Barbillon, Pierre"https://zbmath.org/authors/?q=ai:barbillon.pierre"Chiodetti, Matthieu"https://zbmath.org/authors/?q=ai:chiodetti.matthieu"Keller, Merlin"https://zbmath.org/authors/?q=ai:keller.merlin"Parent, Éric"https://zbmath.org/authors/?q=ai:parent.ericSummary: Field experiments are often difficult and expensive to carry out. To bypass these issues, industrial companies have developed computational codes. These codes are intended to be representative of the physical system, but come with a certain number of problems. Despite continuous code development, the difference between the code outputs and experiments can remain significant. Two kinds of uncertainties are observed. The first one comes from the difference between the physical phenomenon and the values recorded experimentally. The second concerns the gap between the code and the physical system. To reduce this difference, often named model bias, discrepancy, or model error, computer codes are generally complexified in order to make them more realistic. These improvements increase the computational cost of the code. Moreover, a code often depends on user-defined parameters in order to match field data as closely as possible. This estimation task is called calibration. This paper proposes a review of Bayesian calibration methods and is based on an application case which makes it possible to discuss the various methodological choices and to illustrate their divergences. This example is based on a code used to predict the power of a photovoltaic plant.A solution structure-based adaptive approximate (SSAA) Riemann solver for the elastic-perfectly plastic solidhttps://zbmath.org/1516.651112023-09-19T14:22:37.575876Z"Gao, Si"https://zbmath.org/authors/?q=ai:gao.si"Liu, Tiegang"https://zbmath.org/authors/?q=ai:liu.tiegangSummary: The exact Riemann solver for one-dimensional elastic-perfectly plastic solid has been presented in the previous work [S. Gao and T. G. Liu, Adv. Appl. Math. Mech., 9(3), 2017, 621-650], but its iterative process of finding nonlinear equation solution is time-consuming. In this paper, to enhance the computational efficiency of the exact Riemann solver and provide a more practical Riemann solver for actual implementation, we design a non-iterative solution structure-based adaptive approximate (SSAA) Riemann solver for one-dimensional elastic-perfectly plastic solid. Judging the solution structure adaptively and then solving the Riemann problem with corresponding solution structure non-iteratively can shorten the computing time and meanwhile guarantee the correctness of the final result. Numerical performance tests manifest that the exact Riemann solver is indeed time-consuming and the ordinary approximate Riemann solver with fixed three-wave solution structure is of great error, whereas the SSAA Riemann solver is of both efficiency and accuracy. Error estimation further indicates that the SSAA Riemann solver has at least second-order accuracy to approach the exact solution of the states in the star region.Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equationshttps://zbmath.org/1516.651122023-09-19T14:22:37.575876Z"Porteous, William A."https://zbmath.org/authors/?q=ai:porteous.william-a"Laiu, Ming Tse P."https://zbmath.org/authors/?q=ai:laiu.ming-tse-p"Hauck, Cory D."https://zbmath.org/authors/?q=ai:hauck.cory-dSummary: We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and \(C^2\) approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.Differential quadrature method for space-fractional diffusion equations on 2D irregular domainshttps://zbmath.org/1516.651132023-09-19T14:22:37.575876Z"Zhu, X. G."https://zbmath.org/authors/?q=ai:zhu.xiaogang"Yuan, Z. B."https://zbmath.org/authors/?q=ai:yuan.zhanbin"Liu, F."https://zbmath.org/authors/?q=ai:liu.fawang"Nie, Y. F."https://zbmath.org/authors/?q=ai:nie.yufengSummary: In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by Lévy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problemshttps://zbmath.org/1516.651142023-09-19T14:22:37.575876Z"Feng, Qiwei"https://zbmath.org/authors/?q=ai:feng.qiwei"Han, Bin"https://zbmath.org/authors/?q=ai:han.bin|han.bin.1"Minev, Peter"https://zbmath.org/authors/?q=ai:minev.peter-dimitrovSummary: In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.High order compact schemes for flux type BCshttps://zbmath.org/1516.651152023-09-19T14:22:37.575876Z"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilin.1"Pan, Kejia"https://zbmath.org/authors/?q=ai:pan.kejiaThe authors propose an innovative approach for the construction of higher order compact difference schemes which can be applied to general elliptic PDEs with Dirichlet, Robin, or Neumann boundary conditions in two and three dimensions. The idea is to use linear combinations of the solution values, the source term values, and the flux type of BC restricted to the grid points in the compact finite difference stencil. The undetermined coefficients are chosen so that the local truncation errors is as small as desired while maintaining the discrete maximum principle. Therefore, the corresponding coefficient matrices are M-matrices, which guarantee the discrete maximum principle for well-posed problems, and consequently the convergence of the suggested fourth order methods. The numerical experiments with non-trivial test examples which confirm theoretical considerations are given.
Reviewer: Ljiljana Teofanov (Novi Sad)A robust discrete scheme based on staggered grids for poroelastic-elastic coupled problemshttps://zbmath.org/1516.651162023-09-19T14:22:37.575876Z"Yang, Bohan"https://zbmath.org/authors/?q=ai:yang.bohan"Rui, Hongxing"https://zbmath.org/authors/?q=ai:rui.hongxingSummary: This paper aims to explore a robust and efficient discrete scheme for poroelastic-elastic coupled problems. Based on staggered grids, a discrete scheme using finite difference methods is constructed in a form similar to domain decomposition. On this basis, another discrete scheme with a uniform form over the entire domain is derived. These two discrete schemes serve different purposes and are shown to be equivalent. The stability is easily derived by establishing a discrete variational formulation. To prove the convergence, appropriate discrete interpolations are introduced. It is proved that the discrete scheme has second-order superconvergence. Then, combined with the inf-sup condition, the first-order uniform convergence is obtained. This means the discrete scheme has great potential to overcome Poisson locking and pressure oscillations. Some numerical experiments are also carried out, and the results show the robustness and efficiency of the discrete scheme.Construction and analysis of the quadratic finite volume methods on tetrahedral mesheshttps://zbmath.org/1516.651172023-09-19T14:22:37.575876Z"Yang, Peng"https://zbmath.org/authors/?q=ai:yang.peng"Wang, Xiang"https://zbmath.org/authors/?q=ai:wang.xiang.2"Li, Yonghai"https://zbmath.org/authors/?q=ai:li.yonghaiThe authors establish a family of quadratic FVM (Finite Volume Method) schemes on tetrahedral meshes by introducing three parameters on the dual mesh. The method is developed for elliptic equations in 3D. Under the proposed orthogonal conditions and the minimum V-angle condition a theoretical analysis is presented. The stabilty and optimal error estimates in \(H^1\) and \(L^2\) norms are derived and proved. Some numerical tests are presented to confirm the theoretical results.
Reviewer: Abdallah Bradji (Annaba)Quasi-Monte Carlo and multilevel Monte Carlo methods for computing posterior expectations in elliptic inverse problemshttps://zbmath.org/1516.651182023-09-19T14:22:37.575876Z"Scheichl, R."https://zbmath.org/authors/?q=ai:scheichl.robert"Stuart, A. M."https://zbmath.org/authors/?q=ai:stuart.andrew-m"Teckentrup, A. L."https://zbmath.org/authors/?q=ai:teckentrup.aretha-lSummary: We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes's rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo, or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach.Algebraic multigrid using a stencil-CSR hybrid format on GPUshttps://zbmath.org/1516.651192023-09-19T14:22:37.575876Z"Boukhris, Siham"https://zbmath.org/authors/?q=ai:boukhris.siham"Napov, Artem"https://zbmath.org/authors/?q=ai:napov.artem"Notay, Yvan"https://zbmath.org/authors/?q=ai:notay.yvanSummary: We propose a new sparse matrix format which captures the matrix structure typical for discretized partial differential equations with piecewise-constant coefficients. The format uses a stencil representation for some blocks of matrix rows, typically corresponding to regions with constant coefficients, whereas other rows are encoded in the general compressed sparse row format. The stencil representation saves memory and is suitable for SIMD-like parallelism as available on GPUs. Further, this format is well suited for the implementation of algebraic multigrid methods, and we present a proof-of-concept GPU-accelerated aggregation-based algebraic multigrid solver based on this format. This solver is compared on a few model problems (2-dimension and 3-dimension Poisson-like) with the compressed sparse row-based solver AmgX from NVIDIA and with the CUDA version of the BoomerAMG solver. For the considered tested problems with one million unknowns or more, the presented solver outperforms AmgX and BoomerAMG in terms of both run time and memory usage, and the performance gap increases with the system size.Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: conforming approximationshttps://zbmath.org/1516.651202023-09-19T14:22:37.575876Z"Cancès, Eric"https://zbmath.org/authors/?q=ai:cances.eric"Dusson, Geneviève"https://zbmath.org/authors/?q=ai:dusson.genevieve"Maday, Yvon"https://zbmath.org/authors/?q=ai:maday.yvon"Stamm, Benjamin"https://zbmath.org/authors/?q=ai:stamm.benjamin"Vohralík, Martin"https://zbmath.org/authors/?q=ai:vohralik.martinSummary: This paper derives a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue. They are valid without restrictions on the computational mesh or on the approximate eigenvector; we only need to assume that the approximate eigenvalue is separated from the surrounding smaller and larger exact ones, which can be checked in practice. Guaranteed, fully computable, optimally convergent, and polynomial-degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well, under the same hypotheses. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory, and no convexity/regularity assumption on the computational domain/exact eigenvector(s) is needed. The multiplicative constant appearing in our estimates depends on (computable estimates of) the gaps to the surrounding exact eigenvalues. Its two improvements are presented. First, it is reduced by a fixed factor under an explicit, a posteriori calculable condition on the mesh and on the approximate eigenvector-eigenvalue pair. Second, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of one. Inexact algebraic solvers are taken into account; the estimates are valid on each iteration and can serve for the design of adaptive stopping criteria. The application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided, along with a numerical illustration on a set of test problems.Computations of eigenvalues and resonances on perturbed hyperbolic surfaces with cuspshttps://zbmath.org/1516.651212023-09-19T14:22:37.575876Z"Levitin, Michael"https://zbmath.org/authors/?q=ai:levitin.michael-r"Strohmaier, Alexander"https://zbmath.org/authors/?q=ai:strohmaier.alexanderSummary: In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a finite element method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM.Gradient robust mixed methods for nearly incompressible elasticityhttps://zbmath.org/1516.651222023-09-19T14:22:37.575876Z"Basava, Seshadri R."https://zbmath.org/authors/?q=ai:basava.seshadri-r"Wollner, Winnifried"https://zbmath.org/authors/?q=ai:wollner.winnifriedSummary: Within the last years pressure robust methods for the discretization of incompressible fluids have been developed. These methods allow the use of standard finite elements for the solution of the problem while simultaneously removing a spurious pressure influence in the approximation error of the velocity of the fluid, or the displacement of an incompressible solid. To this end, reconstruction operators are utilized mapping discretely divergence free functions to divergence free functions. This work shows that the modifications proposed for Stokes equation by \textit{A. Linke} [Comput. Methods Appl. Mech. Eng. 268, 782--800 (2014; Zbl 1295.76007)] also yield gradient robust methods for nearly incompressible elastic materials without the need to resort to discontinuous finite elements methods as proposed in [\textit{G. Fu} et al., J. Sci. Comput. 86, No. 3, Paper No. 39, 31 p. (2021; Zbl 1460.65146)].Optimal convergence rates in \(L^2\) for a first order system least squares finite element method. I: Homogeneous boundary conditionshttps://zbmath.org/1516.651232023-09-19T14:22:37.575876Z"Bernkopf, Maximilian"https://zbmath.org/authors/?q=ai:bernkopf.maximilian"Melenk, Jens Markus"https://zbmath.org/authors/?q=ai:melenk.jens-markusThe authors consider the Poisson-like second-order model problem written as a system of first-order equations. For the discretization, an \(H(\Omega,\operatorname{div})\times H^1(\Omega)\) conforming least squares formulation is employed. The paper's main contribution is optimal \(L^2(\Omega)\) based convergence result for the least squares approximation \(u_h\) to the scalar variable \(u\). Furthermore, \(hp\) error estimates for the gradient of the scalar variable \(u\) as well as an \(hp\) error estimate for the vector variable \(\phi = -\nabla u\) are obtained in the \(L^2 (\Omega)\) norm. With these optimality results optimal convergence rates are achieved under minimal regularity assumptions on the data. Numerical results are presented to confirm the theoretical results
Reviewer: Bülent Karasözen (Ankara)A HDG method for elliptic problems with integral boundary condition: theory and applicationshttps://zbmath.org/1516.651242023-09-19T14:22:37.575876Z"Bertoluzza, Silvia"https://zbmath.org/authors/?q=ai:bertoluzza.silvia"Guidoboni, Giovanna"https://zbmath.org/authors/?q=ai:guidoboni.giovanna"Hild, Romain"https://zbmath.org/authors/?q=ai:hild.romain"Prada, Daniele"https://zbmath.org/authors/?q=ai:prada.daniele"Prud'homme, Christophe"https://zbmath.org/authors/?q=ai:prudhomme.christophe"Sacco, Riccardo"https://zbmath.org/authors/?q=ai:sacco.riccardo"Sala, Lorenzo"https://zbmath.org/authors/?q=ai:sala.lorenzo"Szopos, Marcela"https://zbmath.org/authors/?q=ai:szopos.marcelaSummary: In this paper, we address the study of elliptic boundary value problems in presence of a boundary condition of integral type (IBC) where the potential is an unknown constant and the flux (the integral of the flux density) over a portion of the boundary is given by a value or a coupling condition. We first motivate our work with realistic examples from nano-electronics, high field magnets and ophthalmology. We then define a general framework stemming from the Hybridizable Discontinuous Galerkin method that accounts naturally for the IBC and we provide a complete analysis at continuous and discrete levels. The implementation in the Feel++ framework is then detailed and the convergence and scalability properties are verified. Finally, numerical experiments performed on the real-life motivating applications are used to illustrate our methodology.Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin finite element methodhttps://zbmath.org/1516.651252023-09-19T14:22:37.575876Z"Bespalov, Alex"https://zbmath.org/authors/?q=ai:bespalov.alexei"Praetorius, Dirk"https://zbmath.org/authors/?q=ai:praetorius.dirk"Ruggeri, Michele"https://zbmath.org/authors/?q=ai:ruggeri.micheleSummary: The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori estimator to control the energy error in multilevel stochastic Galerkin approximations for this class of PDE problems. We prove that the two-level estimator always provides a lower bound for the unknown approximation error, while the upper bound is equivalent to a saturation assumption. We propose and empirically compare three adaptive algorithms, where the structure of the estimator is exploited to perform spatial refinement as well as parametric enrichment. The paper also discusses implementation aspects of computing multilevel stochastic Galerkin approximations.Guaranteed contraction of adaptive inexact \textit{hp}-refinement strategies with realistic stopping criteriahttps://zbmath.org/1516.651262023-09-19T14:22:37.575876Z"Daniel, Patrik"https://zbmath.org/authors/?q=ai:daniel.patrik"Vohralík, Martin"https://zbmath.org/authors/?q=ai:vohralik.martinAuthors' abstract: The purpose of this contribution is to theoretically analyze the adaptive refinement strategies for conforming \(h p\)-finite element approximations of elliptic problems proposed for exact algebraic solvers in [\textit{P. Daniel} et al., Comput. Math. Appl. 76, No. 5, 967--983 (2018; Zbl 1427.65356)] and for inexact algebraic solvers in [\textit{P. Daniel} et al., Comput. Methods Appl. Mech. Eng. 359, Article ID 112607, 30 p. (2020; Zbl 1441.65095)]. Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed \(h p\)-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, the authors have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the \(h p\)-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact \(h p\)-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the \(h\)- and \(p\)-refinements, namely an extension of the marked region, as well as a sufficient \(h\)- or \(p\)-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.
Reviewer: Xiaodi Zhang (Zhengzhou)Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR methodhttps://zbmath.org/1516.651272023-09-19T14:22:37.575876Z"Di Pietro, Daniele A."https://zbmath.org/authors/?q=ai:di-pietro.daniele-antonio"Droniou, Jérôme"https://zbmath.org/authors/?q=ai:droniou.jeromeSummary: In this work we investigate from a broad perspective the reduction of degrees of freedom through serendipity techniques for polytopal methods compatible with Hilbert complexes. We first establish an abstract framework that, given two complexes connected by graded maps, identifies a set of properties enabling the transfer of the homological and analytical properties from one complex to the other. This abstract framework is designed having in mind discrete complexes, with one of them being a reduced version of the other, such as occurring when applying serendipity techniques to numerical methods. We then use this framework as an overarching blueprint to design a serendipity DDR complex. Thanks to the combined use of higher-order reconstructions and serendipity, this complex compares favorably in terms of degrees of freedom (DOF) count to all the other polytopal methods previously introduced and also to finite elements on certain element geometries. The gain resulting from such a reduction in the number of DOFs is numerically evaluated on two model problems: a magnetostatic model, and the Stokes equations.Structure-preserving discretizations of gradient flows for axisymmetric two-phase biomembraneshttps://zbmath.org/1516.651282023-09-19T14:22:37.575876Z"Garcke, Harald"https://zbmath.org/authors/?q=ai:garcke.harald"Nürnberg, Robert"https://zbmath.org/authors/?q=ai:nurnberg.robertThe paper deals with the numerical simulation of the form and evolution of multi-phase biomembranes which is important in order to understand living systems. The authors consider a mathematical model based on a Canham-Helfrich-Evans two-phase elastic energy, which lead to fourth-order geometric evolution problems with highly nonlinear boundary conditions. The problem is discretized by a parametric finite element method in an axisymmetric setting. The presented weak formulation leads to space semi-discretization such that energy decay laws and conservation properties hold. The authors prove these properties and show that the fully space-time discretized schemes are well posed. Finally, several numerical computations demonstrate the experimental order of convergence as well as the ability of the method to compute complex, experimentally observed two-phase biomembranes.
Reviewer: Vit Dolejsi (Praha)Adaptive wavelet methods for elliptic partial differential equations with random operatorshttps://zbmath.org/1516.651292023-09-19T14:22:37.575876Z"Gittelson, Claude Jeffrey"https://zbmath.org/authors/?q=ai:gittelson.claude-jeffreySummary: We apply adaptive wavelet methods to boundary value problems with random coefficients, discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain. Greedy algorithms control the approximate application of the fully discretized random operator, and the construction of sparse approximations to this operator. We suggest a power iteration for estimating errors induced by sparse approximations of linear operators.Finite element analysis of a constrained Dirichlet boundary control problem governed by a linear parabolic equationhttps://zbmath.org/1516.651302023-09-19T14:22:37.575876Z"Gudi, Thirupathi"https://zbmath.org/authors/?q=ai:gudi.thirupathi"Mallik, Gouranga"https://zbmath.org/authors/?q=ai:mallik.gouranga"Sau, Ramesh Ch."https://zbmath.org/authors/?q=ai:sau.ramesh-chThe paper deals with the parabolic Dirichlet boundary control problem
\[
\min J(u, q) = \frac12 \| u- u_d\|^2_{L^2(I;L^2(\Omega ))} + \frac{\lambda}{2} | q|^2_{1,\Omega \times (0,T )},
\]
where \(u\) is the solution of linear heat equation with unit heat conductivity and \(q\) is the boundary datum fulfilling the boundedness constrains. The authors discuss the well-posedness of the problem and derive stability estimates. Moreover, they prove the existence and uniqueness of the solution and derive the optimality system. The first-order necessary optimality condition results in a simplified Signorini-type problem for the control variable. The space discretization of the state variable is done using conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. To discretize the control, conforming prismatic Lagrange finite elements are used. They derive an optimal order of convergence of error in the control, state, and adjoint state under some regularity assumptions on the solutions. The theoretical results are justified by some numerical experiments.
Reviewer: Vit Dolejsi (Praha)An arbitrary-order fully discrete Stokes complex on general polyhedral mesheshttps://zbmath.org/1516.651312023-09-19T14:22:37.575876Z"Hanot, Marien-Lorenzo"https://zbmath.org/authors/?q=ai:hanot.marien-lorenzoAuthor's abstract: In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enrich the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of schemes for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme solving the Stokes equations and validate the convergence rates with various numerical tests.
Reviewer: Xiaodi Zhang (Zhengzhou)New analysis of mixed finite element methods for incompressible magnetohydrodynamicshttps://zbmath.org/1516.651322023-09-19T14:22:37.575876Z"Huang, Yuchen"https://zbmath.org/authors/?q=ai:huang.yuchen"Qiu, Weifeng"https://zbmath.org/authors/?q=ai:qiu.weifeng"Sun, Weiwei"https://zbmath.org/authors/?q=ai:sun.weiweiSummary: This paper focuses on new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space in cooperation with Navier-Stokes equations and the Nédélec's edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec's edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order Nédélec's edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented, which confirm our theoretical analysis.A posteriori optimization of parameters in stabilized methods for convection-diffusion problems. IIhttps://zbmath.org/1516.651332023-09-19T14:22:37.575876Z"John, Volker"https://zbmath.org/authors/?q=ai:john.volker"Knobloch, Petr"https://zbmath.org/authors/?q=ai:knobloch.petr"Wilbrandt, Ulrich"https://zbmath.org/authors/?q=ai:wilbrandt.ulrichSummary: Extensions of algorithms for computing optimal stabilization parameters in finite element methods for convection-diffusion equations are presented. These extensions reduce the dimension of the control space, in comparison to available methods, and thus address the long computing times of these methods. One method is proposed that considers only relevant mesh cells, another method that uses groups of mesh cells, and the combination of both methods is also studied. The incorporation of these methods within a gradient-based optimization procedure, via solving an adjoint problem, is explained. Numerical studies provide impressions on the gain of efficiency as well as on the loss of accuracy if control spaces with reduced dimensions are utilized.
For Part I see [the first author et al., Comput. Methods Appl. Mech. Eng. 200, No. 41--44, 2916--2929 (2011; Zbl 1230.76026)].Trivariate spline collocation methods for numerical solution to 3D Monge-Ampère equationhttps://zbmath.org/1516.651342023-09-19T14:22:37.575876Z"Lai, Ming-Jun"https://zbmath.org/authors/?q=ai:lai.mingjun"Lee, Jinsil"https://zbmath.org/authors/?q=ai:lee.jinsilSummary: We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Ampére equation. Mainly we use the spline collocation method introduced in [\textit{M.-J. Lai} and \textit{J. Lee}, SIAM J. Numer. Anal. 60, No. 5, 2405--2434 (2022; Zbl 1497.65245)] to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.Analysis of an HDG method for the Navier-Stokes equations with Dirac measureshttps://zbmath.org/1516.651352023-09-19T14:22:37.575876Z"Leng, Haitao"https://zbmath.org/authors/?q=ai:leng.haitaoSummary: In two dimensions, we analyze a hybridized discontinuous Galerkin (HDG) method for the Navier-Stokes equations with Dirac measures. The approximate velocity field obtained by the HDG method is shown to be pointwise divergence-free and \(H\)(div)-conforming. Under a smallness assumption on the continuous and discrete solutions, \textit{a posteriori} error estimator, that provides an upper bound for the \(L^2\)-norm in the velocity, is proposed in the convex domain. In the polygonal domain, reliable and efficient \textit{a posteriori} error estimator for the \(W^{1,q}\)-seminorm in the velocity and \(L^q\)-norm in the pressure is also proved. Finally, a Banach's fixed point iteration method and an adaptive HDG algorithm are introduced to solve the discrete system and show the performance of the obtained \textit{a posteriori} error estimators.Well-posedness and finite element approximation for the steady-state closed-loop geothermal systemhttps://zbmath.org/1516.651362023-09-19T14:22:37.575876Z"Liu, Haochen"https://zbmath.org/authors/?q=ai:liu.haochen"Huang, Pengzhan"https://zbmath.org/authors/?q=ai:huang.pengzhan"He, Yinnian"https://zbmath.org/authors/?q=ai:he.yinnianSummary: In this article, we study a multi-physics model for the closed-loop geothermal system consisting of some pipelines and a reservoir. The mathematical model for simulating this system mainly focuses on heat transfer between porous media in the reservoir and fluid flow in the pipelines. Besides, to describe the fluid and heat transfer in two regions, the governing equations of the mathematical model are denoted by the steady Navier-Stokes/Darcy equations coupled with the convection-diffusion equations. Next, we design a perturbation system to prove the well-posedness of the governing equations. Moreover, in variational formulation, a Nitsche's penalty term is added to overcome the difficulty of instability caused by the interface condition of heat flux. Furthermore, a decoupled iterative algorithm is proposed to solve the considered equations based on the finite element method, and the stability and convergence of the presented algorithm are shown. Finally, some numerical tests are performed to verify the accuracy and efficiency of the proposed algorithm.Stability and interpolation properties for Stokes-like virtual element spaceshttps://zbmath.org/1516.651372023-09-19T14:22:37.575876Z"Meng, Jian"https://zbmath.org/authors/?q=ai:meng.jian"Beirão da Veiga, Lourenço"https://zbmath.org/authors/?q=ai:beirao-da-veiga.lourenco"Mascotto, Lorenzo"https://zbmath.org/authors/?q=ai:mascotto.lorenzoSummary: We prove stability bounds for Stokes-like virtual element spaces in two and three dimensions. Such bounds are also instrumental in deriving optimal interpolation estimates. Furthermore, we develop some numerical tests in order to investigate the behaviour of the stability constants also from the practical side.An efficient numerical method based on cubic B-spline for time dependent problem with small parameterhttps://zbmath.org/1516.651382023-09-19T14:22:37.575876Z"Redouane, Kelthoum Lina"https://zbmath.org/authors/?q=ai:redouane.kelthoum-lina"Arar, Nouria"https://zbmath.org/authors/?q=ai:arar.nouria"Al-Mdallal, Qasem"https://zbmath.org/authors/?q=ai:al-mdallal.qasem-mSummary: This work is devoted to the development of a Galerkin-type approximation of the solution of parabolic reaction-diffusion problems, utilizing cubic B-Spline functions and a finite difference scheme. An error estimate for the semi discrete weak Galerkin scheme is established. A Von Neumann stability study of the proposed fully discrete Crank Nicolson scheme is also performed. In addition, examples are used to validate the proposed approximation. The numerical results produced demonstrate the procedure's efficacy and are in good agreement with the exact solution.Weak Galerkin finite element methods for quad-curl problemshttps://zbmath.org/1516.651392023-09-19T14:22:37.575876Z"Wang, Chunmei"https://zbmath.org/authors/?q=ai:wang.chunmei"Wang, Junping"https://zbmath.org/authors/?q=ai:wang.junping"Zhang, Shangyou"https://zbmath.org/authors/?q=ai:zhang.shangyouSummary: This article introduces a weak Galerkin (WG) finite element method for quad-curl problems in three dimensions. It is proved that the proposed WG method is stable and accurate in an optimal order of error estimates for the exact solution in discrete norms. In addition, an \(L^2\) error estimate in an optimal order except the lowest order \(k = 2\) is derived for the WG solution. Some numerical experiments are conducted to verify the efficiency and accuracy of our WG method and furthermore a superconvergence has been observed from the numerical results.Discontinuous Galerkin methods for hemivariational inequalities in contact mechanicshttps://zbmath.org/1516.651402023-09-19T14:22:37.575876Z"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.11"Shah, Sheheryar"https://zbmath.org/authors/?q=ai:shah.sheheryar"Wu, Bangmin"https://zbmath.org/authors/?q=ai:wu.bangminSummary: In this paper, we study discontinuous Galerkin (DG) methods for solving two contact problems. The first problem involves a frictionless normal compliance contact boundary condition, and the second is a bilateral contact problem with friction. These contact problems are modeled by hemivariational inequalities, which consist of non-convex and non-smooth terms. We apply five DG methods to solve the contact problems and establish a priori error estimates for these methods. We prove that the DG schemes achieve optimal convergence order for linear elements. Two examples are presented for numerical evidence of the theoretically predicted convergence order.Superconvergence analysis of curlcurl-conforming elements on rectangular mesheshttps://zbmath.org/1516.651412023-09-19T14:22:37.575876Z"Wang, Lixiu"https://zbmath.org/authors/?q=ai:wang.lixiu"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian.13"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhiminSummary: In our recent work [\textit{K. Hu} et al., SIAM J. Sci. Comput. 42, No. 6, A3859--A3877 (2020; Zbl 1458.65148)], we observed numerically some superconvergence phenomena of the curlcurl-conforming finite elements on rectangular domains. In this paper, we provide a theoretical justification for our numerical observation and establish a superconvergence theory for the curlcurl-conforming elements on rectangular meshes. For the elements with parameters \(r\) (\(r=k-1, k, k+1\)) and \(k\) (\(k\ge 2\)), we show that the first (second) component of the numerical solution \(\boldsymbol{u}_h\) converges with rate \(r+1\) at \(r\) vertical (horizontal) Gaussian lines in each element when \(r=k-1\), \(k\) with \(k\ge 3\), \(\nabla \times \boldsymbol{u}_h\) converges with rate \(k+1\) at \(k^2\) Lobatto points in each element when \(k\ge 3\), and the first (second) component of \(\nabla \times \nabla \times \boldsymbol{u}_h\) converges with rate \(k\) at \((k-1)\) horizontal (vertical) Gaussian lines when \(k\ge 2\). They are all one-order higher than the related optimal rates. More numerical experiments are provided to confirm our theoretical results.Penalty-free any-order weak Galerkin FEMs for linear elasticity on quadrilateral mesheshttps://zbmath.org/1516.651422023-09-19T14:22:37.575876Z"Wang, Ruishu"https://zbmath.org/authors/?q=ai:wang.ruishu"Wang, Zhuoran"https://zbmath.org/authors/?q=ai:wang.zhuoran"Liu, Jiangguo"https://zbmath.org/authors/?q=ai:liu.jiangguoSummary: This paper develops a family of new weak Galerkin (WG) finite element methods (FEMs) for solving linear elasticity in the primal formulation. For a convex quadrilateral mesh, degree \(k \ge 0\) vector-valued polynomials are used independently in element interiors and on edges for approximating the displacement. No penalty or stabilizer is needed for these new methods. The methods are free of Poisson-locking and have optimal order \((k+1)\) convergence rates in displacement, stress, and dilation (divergence of displacement). Numerical experiments on popular test cases are presented to illustrate the theoretical estimates and demonstrate efficiency of these new solvers. Extension to cuboidal hexahedral meshes is briefly discussed.Multi-scale fusion network: a new deep learning structure for elliptic interface problemshttps://zbmath.org/1516.651432023-09-19T14:22:37.575876Z"Ying, Jinyong"https://zbmath.org/authors/?q=ai:ying.jinyong"Liu, Jiaxuan"https://zbmath.org/authors/?q=ai:liu.jiaxuan"Chen, Jiaxin"https://zbmath.org/authors/?q=ai:chen.jiaxin"Cao, Shen"https://zbmath.org/authors/?q=ai:cao.shen"Hou, Muzhou"https://zbmath.org/authors/?q=ai:hou.muzhou"Chen, Yinghao"https://zbmath.org/authors/?q=ai:chen.yinghaoSummary: In this paper, we construct a novel multi-scale fusion network as a new deep learning structure to solve the elliptic interface problem. Compared with the results of the fully connected neural network and ResNet, the new multi-scale fusion network is shown to be able to better capture ``sharp turns'', leading to the improved accuracy. Furthermore, its numerical solutions can preserve the \(C^0\) continuity of the solution while keeping the flux jumps passing through different interfaces, thus maintaining the physics of the differential equation. Then, as an application, the new method \textbf{is} applied to solve the three-dimensional Poisson-Boltzmann equations to calculate the electrostatic potential of immersed biomolecules. Numerical experiments demonstrate the effectiveness of our new method compared to the results obtained by the finite element method.Using the FES framework to derive new physical degrees of freedom for finite element spaces of differential formshttps://zbmath.org/1516.651442023-09-19T14:22:37.575876Z"Zampa, Enrico"https://zbmath.org/authors/?q=ai:zampa.enrico"Alonso Rodríguez, Ana"https://zbmath.org/authors/?q=ai:alonso-rodriguez.ana"Rapetti, Francesca"https://zbmath.org/authors/?q=ai:rapetti.francescaThe paper deals a geometric approach for constructing physical degrees of freedom for sequences of finite element spaces. Within the framework of finite element systems, the author propose new degrees of freedom for the spaces \(\mathcal{P}_r \Lambda^k\) of polynomial differential forms. The symbol \(r\) denotes the level of inter-element subdivision and \(k\) is the dimension of integrals on specific subsimplices of dimension, e.g., circulations for \(k = 1\), fluxes for \(k = 2\) or densities for \(k = n\) (\(n\) is the space dimension). The unisolvance of these finite elements is verified numerically. Particulalry, the numerical results on the Vandermonde matrices corresponding with the case \(k = 1\) and different choices of \(r\ge 1\) are presented for \(n=2\).
Reviewer: Vit Dolejsi (Praha)Joint geometry/frequency analyticity of fields scattered by periodic layered mediahttps://zbmath.org/1516.651452023-09-19T14:22:37.575876Z"Kehoe, Matthew"https://zbmath.org/authors/?q=ai:kehoe.matthew"Nicholls, David P."https://zbmath.org/authors/?q=ai:nicholls.david-pSummary: The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high-order perturbation of surfaces/asymptotic waveform evaluation (HOPS/AWE) methods for numerically simulating scattering returns from periodic diffraction gratings. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a periodic two-layer structure. Furthermore, we establish joint analyticity of these solutions with respect to both geometry and frequency perturbations. This result provides hypotheses under which a rigorous numerical analysis could be conducted on our recently developed HOPS/AWE algorithm.An efficient spectral trust-region deflation method for multiple solutionshttps://zbmath.org/1516.651462023-09-19T14:22:37.575876Z"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.5|li.lin|li.lin.10|li.lin.3|li.lin.2|li.lin.1|li.lin.6"Wang, Li-Lian"https://zbmath.org/authors/?q=ai:wang.lilian"Li, Huiyuan"https://zbmath.org/authors/?q=ai:li.huiyuanSummary: It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.An efficient spectral-Galerkin method for elliptic equations in 2D complex geometrieshttps://zbmath.org/1516.651472023-09-19T14:22:37.575876Z"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqing"Wen, Xian"https://zbmath.org/authors/?q=ai:wen.xian"Yao, Guoqing"https://zbmath.org/authors/?q=ai:yao.guoqingSummary: A polar coordinate transformation is considered, which transforms the complex geometries into a unit disc. Some basic properties of the polar coordinate transformation are given. As applications, we consider the elliptic equation in two-dimensional complex geometries. The existence and uniqueness of the weak solution are proved, the Fourier-Legendre spectral-Galerkin scheme is constructed and the optimal convergence of numerical solutions under \(H^1\)-norm is analyzed. The proposed method is very effective and easy to implement for problems in 2D complex geometries. Numerical results are presented to demonstrate the high accuracy of our spectral-Galerkin method.Symbol based convergence analysis in multigrid methods for saddle point problemshttps://zbmath.org/1516.651482023-09-19T14:22:37.575876Z"Bolten, Matthias"https://zbmath.org/authors/?q=ai:bolten.matthias"Donatelli, Marco"https://zbmath.org/authors/?q=ai:donatelli.marco"Ferrari, Paola"https://zbmath.org/authors/?q=ai:ferrari.paola"Furci, Isabella"https://zbmath.org/authors/?q=ai:furci.isabellaSummary: Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the two-grid method. The present paper analyzes saddle point problems where the blocks are circulant within this framework. It contains sufficient conditions for convergence and optimal parameters for the preconditioning of the unilevel and multilevel saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, it is shown that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and studying the ``level independency'' property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.Structure-preserving algorithms with uniform error bound and long-time energy conservation for highly oscillatory Hamiltonian systemshttps://zbmath.org/1516.651492023-09-19T14:22:37.575876Z"Wang, Bin"https://zbmath.org/authors/?q=ai:wang.bin.21"Jiang, Yaolin"https://zbmath.org/authors/?q=ai:jiang.yaolinSummary: Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear systems with highly oscillatory solution, and the blended algorithms inherit and respect the advantage of each method. Two kinds of algorithms are presented to preserve the symplecticity and energy of the Hamiltonian systems, respectively. Long time energy conservation is analysed for symplectic algorithms and the proposed algorithms are shown to have uniform error bound in the position for the highly oscillatory structure. Moreover, some methods with uniform error bound in the position and in the velocity are derived and analysed. Two numerical experiments are carried out to support all the theoretical results established in this paper by showing the performance of the blended algorithms.On the order of convergence of the iterative Bernstein splines method for fractional order initial value problemshttps://zbmath.org/1516.651502023-09-19T14:22:37.575876Z"Bica, Alexandru Mihai"https://zbmath.org/authors/?q=ai:bica.alexandru-mihaiSummary: In this note we establish the order of convergence of the iterative Bernstein splines method applied to initial value problems for fractional differential equations. The cases corresponding to the fractional order smaller than 2 and bigger than 2 are investigated separately, proving that the iterative Bernstein splines method cannot provide an order of convergence better than 2. Some numerical experiments are finally presented.An efficient spectral-Galerkin method for second kind weakly singular VIEs with highly oscillatory kernelshttps://zbmath.org/1516.651512023-09-19T14:22:37.575876Z"Cai, Haotao"https://zbmath.org/authors/?q=ai:cai.haotaoSummary: In this paper, we construct an efficient spectral Galerkin method to deal with the classical second kind linear VIEs with weakly singular and highly oscillatory kernel. We first study the oscillation and singularity of the exact solution and then based on those results, we propose an efficient fully discrete spectral Galerkin method. The proposed algorithm reaches an optimal convergence order without the influence of the wave number. At last, two numerical examples are provided to verify the efficiency of our proposed method.Fractional collocation method for third-kind Volterra integral equations with nonsmooth solutionshttps://zbmath.org/1516.651522023-09-19T14:22:37.575876Z"Ma, Zheng"https://zbmath.org/authors/?q=ai:ma.zheng"Huang, Chengming"https://zbmath.org/authors/?q=ai:huang.chengmingSummary: In this paper, we develop a collocation method for solving third-kind Volterra integral equations. In order to achieve high order convergence for problems with nonsmooth solutions, we construct a collocation scheme on a modified graded mesh using a basis of fractional polynomials, depending on a certain parameter \(\lambda\). For the proposed method, we derive an error estimate in the \(L^\infty\)-norm, which shows that the optimal order of global convergence can be obtained by choosing the appropriate parameter \(\lambda\) and modified mesh, even when the exact solution has low regularity. Numerical experiments confirm the theoretical results and illustrate the performance of the method.A novel numerical optimality technique to find the optimal results of Volterra integral equation of the second kind with discontinuous kernelhttps://zbmath.org/1516.651532023-09-19T14:22:37.575876Z"Noeiaghdam, Samad"https://zbmath.org/authors/?q=ai:noeiaghdam.samad"Sidorov, Denis"https://zbmath.org/authors/?q=ai:sidorov.denis-nikolaevich"Dreglea, Aliona"https://zbmath.org/authors/?q=ai:dreglea.aliona-iSummary: In this study we consider linear and nonlinear Volterra integral equations (VIEs) of the second kind with discontinuous kernel. A novel iterative method using floating point arithmetic (FPA) is presented to solve the problem. Also a convergence theorem and error analysis of the method are presented. The main novelty of this study is to validate the results using the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method which is based on stochastic arithmetic (SA). Moreover instead of applying usual mathematical softwares we use the CADNA (Control of Accuracy and Debugging for Numerical Applications) library which must be done on Linux operating system using C, C++ or Fortran codes. Applying the method and the library we will be able to find the optimal results such as optimal error, optimal iteration of the method and optimal approximation. Proving the main theorem of the CESTAC method we show the equality between the number of common significant digits (NCSDs) of two successive iterations with the exact and approximate solutions. Thus it can help us to apply a new termination criterion instead of absolute error to show the accuracy and efficiency of the method.Explicit exponential Runge-Kutta methods for semilinear integro-differential equationshttps://zbmath.org/1516.651542023-09-19T14:22:37.575876Z"Ostermann, Alexander"https://zbmath.org/authors/?q=ai:ostermann.alexander"Saedpanah, Fardin"https://zbmath.org/authors/?q=ai:saedpanah.fardin"Vaisi, Nasrin"https://zbmath.org/authors/?q=ai:vaisi.nasrinSummary: The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order \(p\) and prove convergence of order \(p\), whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions, and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.Block Toeplitz inner-bordering method for the Gelfand-Levitan-Marchenko equations associated with the Zakharov-Shabat systemhttps://zbmath.org/1516.651552023-09-19T14:22:37.575876Z"Medvedev, Sergey"https://zbmath.org/authors/?q=ai:medvedev.sergey-v|medvedev.sergei-borisovich"Vaseva, Irina"https://zbmath.org/authors/?q=ai:vaseva.irina-arkadievna"Fedoruk, Mikhail"https://zbmath.org/authors/?q=ai:fedoruk.mikhail-pSummary: We propose a generalized method for solving the Gelfand-Levitan-Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain.On the computation of the SVD of Fourier submatriceshttps://zbmath.org/1516.651562023-09-19T14:22:37.575876Z"Dirckx, S."https://zbmath.org/authors/?q=ai:dirckx.simon"Huybrechs, D."https://zbmath.org/authors/?q=ai:huybrechs.daan"Ongenae, R."https://zbmath.org/authors/?q=ai:ongenae.rSummary: Contiguous submatrices of the Fourier matrix are known to be ill-conditioned. In a recent paper in SIAM review A. \textit{A. H. Barnett} [SIAM Rev. 64, No. 1, 105--131 (2022; Zbl 1505.65325)] has provided new bounds on the rate of ill-conditioning. In this paper we focus on the corresponding singular value decomposition. The singular vectors can be computed from the so-called \textit{periodic discrete prolate spheroidal sequences}, named in analogy to spheroidal wave functions which are associated with the continuous Fourier transform. Their numerical computation is hampered by the clustering of singular values. We collect and expand known results on the stable numerical computation of the singular value decomposition of Fourier submatrices. The prolate sequences are eigenvectors of a tridiagonal matrix whose spectrum is free of clusters and this enables their computation. We collect these observations in a simple and convenient algorithm. The corresponding singular values can be accurately computed as well, except when they are small. Even then, small singular values can be computed in high-precision arithmetic with modest computational effort, even for large and extremely ill-conditioned submatrices. We illustrate the computations and point out a few applications in which Fourier submatrices arise.On Legendre wavelets for Poisson equation in the frame of complex solutionhttps://zbmath.org/1516.651572023-09-19T14:22:37.575876Z"Kerrouche, Nacereddine"https://zbmath.org/authors/?q=ai:kerrouche.nacereddine"Kadem, Abdelouahab"https://zbmath.org/authors/?q=ai:kadem.abdelouahabSummary: In this paper, we study the Poisson equation with a complex solution:
\[
\Delta F (x, y) - \Gamma (x, y) = 0, \quad (x, y) \in [0, 1] \times [0, 1]
\]
\(\Gamma (x, y)\) is a given complex function, \(F(x; y) = f (x; y) + ig(x; y)\) is the unknown complex function, where \(f\) and \(g\) are complex functions twice differentiable on the interval \([0; 1]\), \(\Delta\) denotes the Laplacian operator, \(i\)
is the imaginary unit. The method is based on Legendre wavelets (LW), and the idea, is that the two integration and derivation matrices are mutually used to reduce the problem in order to study numerically linear algebraic system. Some illustrative example are presented to explain the efficiency and simplicity of the presented method.Development of a parallel CUDA algorithm for solving 3D guiding center problemshttps://zbmath.org/1516.651582023-09-19T14:22:37.575876Z"Bak, Soyoon"https://zbmath.org/authors/?q=ai:bak.soyoon"Kim, Philsu"https://zbmath.org/authors/?q=ai:kim.philsu"Park, Sangbeom"https://zbmath.org/authors/?q=ai:park.sangbeomSummary: In this study, we develop a novel compute unified device architecture (CUDA) algorithm, which we call C-ECM3, for solving a three-dimensional (3D) guiding center problem. The C-ECM3 is a parallel algorithm for the iterative-free backward semi-Lagrangian method with third-order temporal accuracy (ECM3). One well known challenge in speeding up a CUDA program is to efficiently design kernel functions that can optimally use hierarchical memory classified according to access speed. To solve this challenge, the C-ECM3 is mainly devoted to making a decomposition strategy for solving the tremendous number of generated Cauchy problems. The decomposition strategy divides the \(9 \times 9\) linear system for each Cauchy problem in the ECM3 into two \(3 \times 3\) linear systems, more solverable parts. In addition, the strategy explicitly solves these small systems using Cramer's rule. It turns out that the proposed C-ECM3 enables us to design an array-free kernel function that efficiently uses hierarchical memory. In addition, the C-ECM3 significantly reduces the run-time for tracing trajectories of particles compared to other graphics processing unit (GPU) programs that use the usual Gaussian algorithm. The Kelvin-Helmholtz instability and a 3D guiding center problem are simulated to demonstrate the numerical evidence for the C-ECM3. With these numerical experiments, we verify that the proposed C-ECM3 significantly improves computational speed compared to other methods while maintaining the accuracy of the CPU (central processing unit) version of ECM3. The validity of the C-ECM3 is also confirmed by showing that it satisfies Shoucri's analysis for Kelvin-Helmholtz instability.Parallel numerical Picard iteration methodshttps://zbmath.org/1516.651592023-09-19T14:22:37.575876Z"Wang, Yinkun"https://zbmath.org/authors/?q=ai:wang.yinkunSummary: In this work, we propose a new class of parallel time integrators for initial-value problems based on the well-known Picard iteration. To this end, we first investigate a class of sequential integrators, known as numerical Picard iteration methods, which falls into the general framework of deferred correction methods. We show that the numerical Picard iteration methods admit a \(\min (J,M+1)\)-order rate of convergence, where \(J\) denotes the number of Picard iterations and \(M+1\) is the number of collocation points. We then propose a class of parallel solvers so that \(J\) Picard iterations can be proceeded simultaneously and nearly constantly. We show that the parallel solvers yield the same convergence rate as that of the numerical Picard iteration methods. The main features of the proposed parallelized approach are as follows. (1) Instead of computing the solution point by point [as in revisionist integral deferred correction (RIDC) methods], the proposed methods proceed segment by segment. (2) The proposed approach leads to a higher speedup; the speedup is shown to be \(J(M+1)\) (while the speedup of the \(J\)th order RIDC is, at most, \(J)\). (3) The approach is applicable for non-uniform points, such as Chebyshev points. The stability region of the proposed methods is analyzed in detail, and we present numerical examples to verify the theoretical findings.NFFT.jl: generic and fast Julia implementation of the nonequidistant fast Fourier transformhttps://zbmath.org/1516.651602023-09-19T14:22:37.575876Z"Knopp, Tobias"https://zbmath.org/authors/?q=ai:knopp.tobias"Boberg, Marija"https://zbmath.org/authors/?q=ai:boberg.marija"Grosser, Mirco"https://zbmath.org/authors/?q=ai:grosser.mircoSummary: The nonequidistant fast Fourier transform (NFFT) is an extension of the famous fast Fourier transform (FFT) that can be applied to nonequidistantly sampled data in time/space or frequency domain. It is an approximative algorithm that allows one to control the approximation error in such a way that machine precision is reached while keeping the algorithmic complexity in the same order as a regular FFT. The NFFT plays a major role in many signal processing applications and has been intensively studied from a theoretical and computational perspective. The fastest CPU implementations of the NFFT are implemented in the low-level programming languages C and C++ and require a compromise between code generalizability, code readability, and code efficiency. The programming language Julia promises new opportunities in optimizing these three conflicting goals. In this work we show that Julia indeed allows one to develop an NFFT implementation which is completely generic and dimension-agnostic and requires about two to three times less code than the other famous libraries NFFT3 and FINUFFT while still being one of the fastest NFFT implementations developed to date.\texttt{Procrustes}: a Python library to find transformations that maximize the similarity between matriceshttps://zbmath.org/1516.651612023-09-19T14:22:37.575876Z"Meng, Fanwang"https://zbmath.org/authors/?q=ai:meng.fanwang"Richer, Michael"https://zbmath.org/authors/?q=ai:richer.michael"Tehrani, Alireza"https://zbmath.org/authors/?q=ai:tehrani.alireza-mosleh"La, Jonathan"https://zbmath.org/authors/?q=ai:la.jonathan"Kim, Taewon David"https://zbmath.org/authors/?q=ai:kim.taewon-david"Ayers, Paul W."https://zbmath.org/authors/?q=ai:ayers.paul-w"Heidar-Zadeh, Farnaz"https://zbmath.org/authors/?q=ai:zadeh.farnaz-heidarSummary: We have developed \texttt{Procrustes}, a free, open-source, cross-platform, and user-friendly Python library implementing a wide-range of algorithmic solutions to Procrustes problems. The goal of Procrustes analysis is to find an optimal transformation that makes two matrices as close as possible to each other, where the matrices are often (but need not always be) a list of multidimensional points specifying the systems of interest. We demonstrate the functionality of the package through various examples, mostly from cheminformatics. However, Procrustes analysis has broad applicability including image recognition, signal processing, data science, machine learning, computational biology, chemistry, and physics. Our library includes methods for one-sided Procrustes problems using orthogonal, rotational, symmetric, and permutation transformation matrices, as well as two-sided Procrustes problems using orthogonal and permutation transformation matrices. For the two-sided permutation Procrustes problem, we include heuristic algorithms along with a rigorous (but slow) method based on softassign. In addition, we include a general formulation of the Procrustes problem. The \texttt{Procrustes} source code and documentation is hosted on GitHub (\url{https://github.com/theochem/procrustes}).The software design of gridap: a finite element package based on the Julia JIT compilerhttps://zbmath.org/1516.651622023-09-19T14:22:37.575876Z"Verdugo, Francesc"https://zbmath.org/authors/?q=ai:verdugo.francesc"Badia, Santiago"https://zbmath.org/authors/?q=ai:badia.santiagoSummary: We present the software design of Gridap, a novel finite element library written exclusively in the Julia programming language, which is being used by several research groups world-wide to simulate complex physical phenomena such as magnetohydrodynamics, photonics, weather modeling, non-linear solid mechanics, and fluid-structure interaction problems. The library provides a feature-rich set of discretization techniques for the numerical approximation of a wide range of mathematical models governed by Partial Differential Equations (PDEs), including linear, nonlinear, single-field, and multi-field equations. An expressive API allows users to define PDEs in weak form by a syntax close to the mathematical notation. While this is also available in previous frameworks, the main novelty of Gridap is that it implements this API without introducing a domain-specific language plus a compiler of variational forms. Instead, it leverages the Julia just-in-time compiler to build efficient code, specialized for the concrete problem at hand. As a result, there is no need to use different languages for the computational back-end and the user front-end anymore, thus eliminating the so-called two-language problem. Gridap also provides a low-level API that is modular and extensible via the multiple-dispatch paradigm of Julia and provides easy access to the main building blocks of the library if required. The main contribution of this paper is the detailed presentation of the novel software abstractions behind the Gridap design that leverages the new software possibilities provided by the Julia language. The second main contribution of the article is a performance comparison against FEniCS. We measure CPU times needed to assemble discrete systems of linear equations for different problem types and show that the performance of Gridap is comparable to FEniCS, demonstrating that the new software design does not compromise performance. Gridap is freely available at Github (\url{https://github.com/gridap/Gridap.jl}) and distributed under an MIT license.Learning adaptive regularization for image labeling using geometric assignmenthttps://zbmath.org/1516.680802023-09-19T14:22:37.575876Z"Hühnerbein, Ruben"https://zbmath.org/authors/?q=ai:huhnerbein.ruben"Savarino, Fabrizio"https://zbmath.org/authors/?q=ai:savarino.fabrizio"Petra, Stefania"https://zbmath.org/authors/?q=ai:petra.stefania"Schnörr, Christoph"https://zbmath.org/authors/?q=ai:schnorr.christophSummary: We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determines the regularization properties of the assignment flow. Using the symplectic partitioned Runge-Kutta method for numerical integration, it is shown that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A convenient property of our approach is that learning is based on exact inference. Carefully designed experiments demonstrate the performance of our approach, the expressiveness of the mathematical model as well as its limitations, from the viewpoint of statistical learning and optimal control.An elastica-driven digital curve evolution model for image segmentationhttps://zbmath.org/1516.681112023-09-19T14:22:37.575876Z"Antunes, Daniel"https://zbmath.org/authors/?q=ai:antunes.daniel-martins"Lachaud, Jacques-Olivier"https://zbmath.org/authors/?q=ai:lachaud.jacques-olivier"Talbot, Hugues"https://zbmath.org/authors/?q=ai:talbot.huguesSummary: Geometric priors have been shown to be useful in image segmentation to regularize the results. For example, the classical Mumford-Shah functional uses region perimeter as prior. This has inspired much research in the last few decades, with classical approaches like the Rudin-Osher-Fatemi and most graph-cut formulations, which all use a weighted or binary perimeter prior. It has been observed that this prior is not suitable in many applications, for example for segmenting thin objects or some textures, which may have high perimeter/surface ratio. Mumford observed that an interesting prior for natural objects is the Euler elastical model, which involves the squared curvature. In other areas of science, researchers have noticed that some physical binarization processes, like emulsion unmixing, can be well-approximated by curvature-related flow like the Willmore flow. However, curvature-related flows are not easy to compute because curvature is difficult to estimate accurately, and the underlying optimization processes are not convex. In this article, we propose to formulate a digital flow that approximates an Elastica-related flow using a multigrid-convergent curvature estimator, within a discrete variational framework. We also present an application of this model as a post-processing step to a segmentation framework.Effective two-stage image segmentation: a new non-Lipschitz decomposition approach with convergent algorithmhttps://zbmath.org/1516.681122023-09-19T14:22:37.575876Z"Guo, Xueyan"https://zbmath.org/authors/?q=ai:guo.xueyan"Xue, Yunhua"https://zbmath.org/authors/?q=ai:xue.yunhua"Wu, Chunlin"https://zbmath.org/authors/?q=ai:wu.chunlinSummary: Image segmentation is an important median level vision topic. Accurate and efficient multiphase segmentation for images with intensity inhomogeneity is still a great challenge. We present a new two-stage multiphase segmentation method trying to tackle this, where the key is to compute an inhomogeneity-free approximate image. For this, we propose to use a new non-Lipschitz variational decomposition model in the first stage. The minimization problem is solved by an iterative support shrinking algorithm. By assuming that the subproblem at each iteration is exactly solved, we show the global convergence of the iterative algorithm and a lower bound theory of the image gradient of the iterative sequence, which indicates that the generated approximate image (inhomogeneity-corrected component) is with very neat edges and suitable for the following thresholding operation. Implementation details based on the alternating direction method of multipliers for the strongly convex subproblems are also given. In the second stage, the segmentation is done by applying a widely used simple thresholding technique to the piecewise constant approximation. Numerical experiments indicate good convergence properties and effectiveness of our method in multiphase segmentation for either clean or noisy homogeneous and inhomogeneous images. Both visual and quantitative comparisons with some state-of-the-art approaches demonstrate the performance advantages of our non-Lipschitz-based method.Simultaneous image enhancement and restoration with non-convex total variationhttps://zbmath.org/1516.681132023-09-19T14:22:37.575876Z"Kang, Myeongmin"https://zbmath.org/authors/?q=ai:kang.myeongmin"Jung, Miyoun"https://zbmath.org/authors/?q=ai:jung.miyounSummary: In this article, we propose a novel variational model for the joint enhancement and restoration of low-light images corrupted by blurring and/or noise. The model decomposes a given low-light image into reflectance and illumination images that are recovered from blurring and/or noise. In addition, our approach utilizes non-convex total variation regularization on all variables. This allows us to adequately denoise homogeneous regions while preserving the details and edges in both reflectance and illumination images, which leads to clean and sharp final enhanced images. To solve the non-convex model, we employ a proximal alternating minimization approach, and then an iteratively reweighted \(\ell_1\) algorithm and an alternating direction method of multipliers are adopted for solving the subproblems. These techniques contribute to an efficient iterative algorithm, with its convergence proven. Experimental results demonstrate the effectiveness of the proposed model when compared to other state-of-the-art methods in terms of both visual aspect and image quality measures.Predictive online optimisation with applications to optical flowhttps://zbmath.org/1516.681162023-09-19T14:22:37.575876Z"Valkonen, Tuomo"https://zbmath.org/authors/?q=ai:valkonen.tuomoSummary: Online optimisation revolves around new data being introduced into a problem while it is still being solved; think of deep learning as more training samples become available. We adapt the idea to dynamic inverse problems such as video processing with optical flow. We introduce a corresponding \textit{predictive online primal-dual proximal splitting} method. The video frames now \textit{exactly correspond to the algorithm iterations}. A user-prescribed predictor describes the evolution of the primal variable. To prove convergence we need a predictor for the dual variable based on (proximal) gradient flow. This affects the model that the method asymptotically minimises. We show that for inverse problems the effect is, essentially, to construct a new dynamic regulariser based on infimal convolution of the static regularisers with the temporal coupling. We finish by demonstrating excellent real-time performance of our method in computational image stabilisation and convergence in terms of regularisation theory.Learning port-Hamiltonian systems -- algorithmshttps://zbmath.org/1516.700202023-09-19T14:22:37.575876Z"Salnikov, V."https://zbmath.org/authors/?q=ai:salnikov.vladimir|salnikov.vsevolod|salnikov.valeriy|salnikov.v-n|salnikov.v-d"Falaize, A."https://zbmath.org/authors/?q=ai:falaize.antoine"Lozienko, D."https://zbmath.org/authors/?q=ai:lozienko.dSummary: In this article we study the possibilities of recovering the structure of port-Hamiltonian systems starting from ``unlabelled'' ordinary differential equations describing mechanical systems. The algorithm we suggest solves the problem in two phases. It starts by constructing the connectivity structure of the system using machine learning methods -- producing thus a graph of interconnected subsystems. Then this graph is enhanced by recovering the Hamiltonian structure of each subsystem as well as the corresponding ports. This second phase relies heavily on results from symplectic and Poisson geometry that we briefly sketch. And the precise solutions can be constructed using methods of computer algebra and symbolic computations. The algorithm permits to extend the port-Hamiltonian formalism to generic ordinary differential equations, hence introducing eventually a new concept of normal forms of ODEs.Special issue: 92nd annual meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Aachen, Germany, virtual, August 15--19, 2022https://zbmath.org/1516.740022023-09-19T14:22:37.575876ZThe articles of this volume will not be indexed individually. For the preceding conference see [Zbl 1494.74001].Special issue: 92nd annual meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Aachen, Germany, virtual, August 15--19, 2022https://zbmath.org/1516.740032023-09-19T14:22:37.575876ZThe articles of this volume will not be indexed individually. For the preceding conference see [Zbl 1494.74001].On effects of concentrated loads on perforated sensitive shells of revolutionhttps://zbmath.org/1516.741002023-09-19T14:22:37.575876Z"Giani, Stefano"https://zbmath.org/authors/?q=ai:giani.stefano"Hakula, Harri"https://zbmath.org/authors/?q=ai:hakula.harriSummary: Sensitive shells are a class of constructions with specific and perhaps unintuitive responses to different loading scenarios. There are new emerging applications for capsules with open cavities, and as the designs are optimised to maximise the payload, sensitivity has to be taken into account. Concentrated loads induce singularities that propagate along the characteristics of the surfaces. Perforation patterns do not have a strong effect on internal layers. Under a symmetric concentrated loads for parabolic and hyperbolic shells there exists a critical thickness at which local features of solution begin to dominate, whereas for elliptic cases there is always a dominant global response. In non-uniform curvature cases the elliptic part will dictate the solution even if the load is not acting on it. The extensive set of simulations has been computed using high-order finite element solvers including adaptivity.LAMMPS -- a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scaleshttps://zbmath.org/1516.741082023-09-19T14:22:37.575876Z"Thompson, Aidan P."https://zbmath.org/authors/?q=ai:thompson.aidan-p"Aktulga, H. Metin"https://zbmath.org/authors/?q=ai:aktulga.hasan-metin"Berger, Richard"https://zbmath.org/authors/?q=ai:berger.richard"Bolintineanu, Dan S."https://zbmath.org/authors/?q=ai:bolintineanu.dan-s"Brown, W. Michael"https://zbmath.org/authors/?q=ai:brown.w-michael"Crozier, Paul S."https://zbmath.org/authors/?q=ai:crozier.paul-s"in 't Veld, Pieter J."https://zbmath.org/authors/?q=ai:in-t-veld.pieter-j"Kohlmeyer, Axel"https://zbmath.org/authors/?q=ai:kohlmeyer.axel"Moore, Stan G."https://zbmath.org/authors/?q=ai:moore.stan-g"Nguyen, Trung Dac"https://zbmath.org/authors/?q=ai:nguyen.trung-dac"Shan, Ray"https://zbmath.org/authors/?q=ai:shan.ray"Stevens, Mark J."https://zbmath.org/authors/?q=ai:stevens.mark-j"Tranchida, Julien"https://zbmath.org/authors/?q=ai:tranchida.julien"Trott, Christian"https://zbmath.org/authors/?q=ai:trott.christian"Plimpton, Steven J."https://zbmath.org/authors/?q=ai:plimpton.steven-jSummary: Since the classical molecular dynamics simulator LAMMPS was released as an open source code in 2004, it has become a widely-used tool for particle-based modeling of materials at length scales ranging from atomic to mesoscale to continuum. Reasons for its popularity are that it provides a wide variety of particle interaction models for different materials, that it runs on any platform from a single CPU core to the largest supercomputers with accelerators, and that it gives users control over simulation details, either via the input script or by adding code for new interatomic potentials, constraints, diagnostics, or other features needed for their models. As a result, hundreds of people have contributed new capabilities to LAMMPS and it has grown from fifty thousand lines of code in 2004 to a million lines today. In this paper several of the fundamental algorithms used in LAMMPS are described along with the design strategies which have made it flexible for both users and developers. We also highlight some capabilities recently added to the code which were enabled by this flexibility, including dynamic load balancing, on-the-fly visualization, magnetic spin dynamics models, and quantum-accuracy machine learning interatomic potentials.Heterogeneous multiscale methods for rough-wall laminar viscous flowhttps://zbmath.org/1516.760512023-09-19T14:22:37.575876Z"Carney, Sean P."https://zbmath.org/authors/?q=ai:carney.sean-p"Engquist, Björn"https://zbmath.org/authors/?q=ai:engquist.bjorn-eSummary: We develop numerical multiscale methods for viscous fluid flow over a rough boundary. The goal is to derive effective boundary conditions, or wall laws, through high resolution simulations localized to the boundary coupled to a coarser simulation in the domain interior following the framework of the heterogeneous multiscale method. Rigorous convergence of the coupled system is shown in a simplified setting. Numerical experiments illustrate the utility of the method for more general roughness patterns and far field flow conditions.multiUQ: a software package for uncertainty quantification of multiphase flowshttps://zbmath.org/1516.760642023-09-19T14:22:37.575876Z"Turnquist, Brian"https://zbmath.org/authors/?q=ai:turnquist.brian-p"Owkes, Mark"https://zbmath.org/authors/?q=ai:owkes.markSummary: multiUQ is a novel tool that simulates gas-liquid multiphase flows and quantifies uncertainty in results due to variability about fluid properties and initial/boundary conditions. The benefit over a typical deterministic solver is that inexact information, such as variability in fluid properties or flow rates, can be included to determine the affect on simulation solutions. It is common to deploy \textit{non-intrusive} methods which utilize many solutions from a deterministic solver to generate a distribution of possible results. Contrarily, multiUQ uses an \textit{intrusive} uncertainty quantification method wherein variables of interest are functions of space, time, and additional uncertainty dimensions. The intrusive solver is run once, giving a distribution of solutions as an output, as well as desired statistics. We use polynomial chaos to create the stochastic variables, which represent a distribution of values at each grid point. The stochastic variables are substituted into the incompressible Navier-Stokes equations, which govern the stochastic fluid dynamics. A stochastic level set is used to capture the distribution of interfaces that are present in an uncertain multiphase flow. multiUQ is written in Fortran and uses a message passing interface (MPI) for parallel operation. Given the many applications of multiphase flows, including open flows, hydraulics, fuel injection systems, and atomizing jets, there is a massive potential benefit to calculating uncertainty information about these flows in a cost-effective manner.Uncertainty quantification for mineral precipitation and dissolution in fractured porous mediahttps://zbmath.org/1516.760792023-09-19T14:22:37.575876Z"Botti, Michele"https://zbmath.org/authors/?q=ai:botti.michele"Fumagalli, Alessio"https://zbmath.org/authors/?q=ai:fumagalli.alessio"Scotti, Anna"https://zbmath.org/authors/?q=ai:scotti.annaSummary: In this work we present an uncertainty quantification analysis to determine the influence and importance of some physical parameters in a reactive transport model in fractured porous media. An accurate description of flow and transport in the fractures is key to obtain reliable simulations, however, fractures geometry and physical characteristics pose several challenges from both the modeling and implementation side. We adopt a mixed-dimensional approximation, where fractures and their intersections are represented as objects of lower dimension. To simplify the presentation, we consider only two chemical species: one solute, transported by water, and one precipitate attached to the solid skeleton. A global sensitivity analysis to uncertain input data is performed exploiting the Polynomial Chaos expansion along with spectral projection methods on sparse grids.Corrigendum to: ``Fixed-grid method for the modelling of unsteady partial oxidation of a spherical coal particle''https://zbmath.org/1516.800242023-09-19T14:22:37.575876Z"Safronov, Dmitry"https://zbmath.org/authors/?q=ai:safronov.dmitry"Nikrityuk, Petr"https://zbmath.org/authors/?q=ai:nikrityuk.petr-a"Meyer, Bernd"https://zbmath.org/authors/?q=ai:meyer.bernd-eCaption for Figure 4 and image and caption for Figure 5 in the authors' paper [ibid. 16, No. 4, 589--610 (2012; Zbl 1516.80023)] are corrected.Propagation of premixed laminar flames in 3D narrow open ducts using RBF-generated finite differenceshttps://zbmath.org/1516.800282023-09-19T14:22:37.575876Z"Bayona, Victor"https://zbmath.org/authors/?q=ai:bayona.victor"Kindelan, Manuel"https://zbmath.org/authors/?q=ai:kindelan.manuel-segura(no abstract)Scalable quantum computation based on nitrogen-vacancy centers in decoherence-free subspacehttps://zbmath.org/1516.810662023-09-19T14:22:37.575876Z"You, Yi"https://zbmath.org/authors/?q=ai:you.yi"Ding, Zhong"https://zbmath.org/authors/?q=ai:ding.zhong"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.19|zhang.yong.28|zhang.yong|zhang.yong.8|zhang.yong.13|zhang.yong.12|zhang.yong.10|zhang.yong.49|zhang.yong.5|zhang.yong.9|zhang.yong.14|zhang.yong.2|zhang.yong.21|zhang.yong.15|zhang.yong.18|zhang.yong.4Summary: Due to its unique optical properties, nitrogen-vacancy centers in diamond show remarkable advantages in realizing quantum information processing and computation. This paper proposes a scalable quantum computing architecture based on solid-state NV centers. In our scheme, logical qubits are encoded in a decoherence-free subspace (DFS) with Larmor pairs (a pair of the nucleus). And the connection between multiple qubits is assisted by a cantilever probe. Then the high fidelity of the universal quantum gate is achieved by using a series of pulses. Our scheme provides physical feasibility for scalable quantum computing and may pave the way for large-scale quantum computing based on NV centers.Quantum image encryption based on block geometric and Haar wavelet transformhttps://zbmath.org/1516.810702023-09-19T14:22:37.575876Z"Fan, Ping"https://zbmath.org/authors/?q=ai:fan.ping"Hou, MengJuan"https://zbmath.org/authors/?q=ai:hou.mengjuan"Hu, WenWen"https://zbmath.org/authors/?q=ai:hu.wenwen"Xiao, Ke"https://zbmath.org/authors/?q=ai:xiao.keSummary: Combine the block-based geometric transformation and Haar wavelet transform, this paper proposes a quantum image encryption for the flexible representation of quantum images. To encrypt the quantum image, the present algorithm mainly contains two stages: first, the quantum block-based geometric transformation performed to disturb the pixels coordinate information iteratively in spatial domain; second, based on 1st quantum Haar wavelet transform, the quantum block-based geometric transformation performed to disturb the approximation information of quantum image in frequency domain. Thereafter, the inverse 1st quantum Haar wavelet transform performed to obtain the final encrypted quantum image. The circuit complexity that realizes the present quantum image encryption and decryption process is only \(\mathrm{O}\left({n}^2\right)\) for a \(2^n\times 2^n\) quantum image, which demonstrates greatly speed-up than classical image encryption algorithms. The experimental results simulated on a classical computer with MATLAB environments indicate that the present algorithm has a good performance for image encryption.Comment on ``PDM Klein-Gordon oscillators in cosmic string spacetime in magnetic and Aharonov-Bohm flux fields within the Kaluza-Klein theory''https://zbmath.org/1516.810782023-09-19T14:22:37.575876Z"Fernández, Francisco M."https://zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: We show that the spectra of some models with supposed physical utility derived recently are not correct. The eigenvalues to the radial differential equations do not satisfy well-known theorems. The origin of the problem is that those equations are not exactly solvable but conditionally solvable. Present analysis shows that the author overlooked a second condition that provides a relationship between the model parameters. Consequently, all the conclusions derived from those supposedly exact eigenvalues cannot be correct.Meta-Schrödinger invariancehttps://zbmath.org/1516.811152023-09-19T14:22:37.575876Z"Stoimenov, Stoimen"https://zbmath.org/authors/?q=ai:stoimenov.stoimen"Henkel, Malte"https://zbmath.org/authors/?q=ai:henkel.malteSummary: The Meta-Schrödinger algebra arises as the dynamical symmetry in transport processes which are ballistic in a chosen `parallel' direction and diffusive in all other `transverse' directions. The time-space transformations of this Lie algebra and its infinite-dimensional extension, the meta-Schrödinger-Virasoro algebra, are constructed. We also find the representations suitable for non-stationary systems by proposing a generalised form of the generator of time-translations. Co-variant two-point functions of quasi-primary scaling operators are derived for both the stationary and the non-stationary cases.Selected topics in analytic conformal bootstrap: a guided journeyhttps://zbmath.org/1516.811652023-09-19T14:22:37.575876Z"Bissi, Agnese"https://zbmath.org/authors/?q=ai:bissi.agnese"Sinha, Aninda"https://zbmath.org/authors/?q=ai:sinha.aninda"Zhou, Xinan"https://zbmath.org/authors/?q=ai:zhou.xinanSummary: This review aims to offer a pedagogical introduction to the analytic conformal bootstrap program via a journey through selected topics. We review analytic methods which include the large spin perturbation theory, Mellin space methods and the Lorentzian inversion formula. These techniques are applied to a variety of topics ranging from large-\(N\) theories, to the epsilon expansion and holographic superconformal correlators, and are demonstrated in a large number of explicit examples.Finite difference time domain simulation of arbitrary shapes quantum dotshttps://zbmath.org/1516.811982023-09-19T14:22:37.575876Z"Parto, Elyas"https://zbmath.org/authors/?q=ai:parto.elyas"Rezaei, Ghasem"https://zbmath.org/authors/?q=ai:rezaei.ghasem"Eslami, Ahmad Mohammadi"https://zbmath.org/authors/?q=ai:eslami.ahmad-mohammadi"Jalali, Tahmineh"https://zbmath.org/authors/?q=ai:jalali.tahminehSummary: Utilizing the finite difference time domain (FDTD) method, energy eigenvalues of spherical, cylindrical, pyramidal and cone-like quantum dots are calculated. To do this, by the imaginary time transformation, we transform the schrödinger equation into a diffusion equation. Then, the FDTD algorithm is hired to solve this equation. We calculate four lowest energy eigenvalues of these QDs and then compared the simulation results with analytical ones. Our results clearly show that simulation results are in very good agreement with analytical results. Therefore, we can use the FDTD method to find accurate results for the Schrödinger equation.A straight forward path to a path integration of Einstein's gravityhttps://zbmath.org/1516.830212023-09-19T14:22:37.575876Z"Klauder, John R."https://zbmath.org/authors/?q=ai:klauder.john-rSummary: Path integration is a respected form of quantization that all theoretical quantum physicists should welcome. This elaboration begins with simple examples of three different versions of path integration. After an important clarification of how gravity can be properly quantized, an appropriate path integral, that also incorporates necessary constraint issues, becomes a proper path integral for gravity that can effectively be obtained. How to evaluate such path integrals is another aspect, but most likely best done by computational efforts including Monte Carlo-like procedures.A projected Nesterov-Kaczmarz approach to stellar population-kinematic distribution reconstruction in extragalactic archaeologyhttps://zbmath.org/1516.850012023-09-19T14:22:37.575876Z"Hinterer, Fabian"https://zbmath.org/authors/?q=ai:hinterer.fabian"Hubmer, Simon"https://zbmath.org/authors/?q=ai:hubmer.simon"Jethwa, Prashin"https://zbmath.org/authors/?q=ai:jethwa.prashin"Soodhalter, Kirk M."https://zbmath.org/authors/?q=ai:soodhalter.kirk-m"van de Ven, Glenn"https://zbmath.org/authors/?q=ai:van-de-ven.glenn"Ramlau, Ronny"https://zbmath.org/authors/?q=ai:ramlau.ronnySummary: In this paper, we consider the problem of reconstructing a galaxy's stellar population-kinematic distribution function from optical integral field unit measurements. These quantities are connected via a high-dimensional integral equation. To solve this problem, we propose a projected Nesterov-Kaczmarz reconstruction method, which efficiently leverages the problem structure and incorporates physical prior information such as smoothness and nonnegativity constraints. To test the performance of our reconstruction approach, we apply it to a dataset simulated from a known ground truth density, and validate it by comparing our recoveries to those obtained by the widely used pPXF software.Learning dynamical systems with side informationhttps://zbmath.org/1516.900422023-09-19T14:22:37.575876Z"Ahmadi, Amir Ali"https://zbmath.org/authors/?q=ai:ahmadi.amir-ali"Khadir, Bachir El"https://zbmath.org/authors/?q=ai:el-khadir.bachirSummary: We present a mathematical and computational framework for learning a dynamical system from noisy observations of a few trajectories and subject to \textit{side information}. Side information is any knowledge we might have about the dynamical system we would like to learn, besides trajectory data, and is typically inferred from domain-specific knowledge or basic principles of a scientific discipline. We are interested in explicitly integrating side information into the learning process in order to compensate for scarcity of trajectory observations. We identify six types of side information that arise naturally in many applications and lead to convex constraints in the learning problem. First, we show that when our model for the unknown dynamical system is parameterized as a polynomial, we can impose our side information constraints computationally via semidefinite programming. We then demonstrate the added value of side information for learning the dynamics of basic models in physics and cell biology, as well as for learning and controlling the dynamics of a model in epidemiology. Finally, we study how well polynomial dynamical systems can approximate continuously differentiable ones while satisfying side information (either exactly or approximately). Our overall learning methodology combines ideas from convex optimization, real algebra, dynamical systems, and functional approximation theory, and can potentially lead to new synergies among these areas.Loss functions for finite setshttps://zbmath.org/1516.900442023-09-19T14:22:37.575876Z"Nie, Jiawang"https://zbmath.org/authors/?q=ai:nie.jiawang"Zhong, Suhan"https://zbmath.org/authors/?q=ai:zhong.suhanSummary: This paper studies loss functions for finite sets. For a given finite set \(S\), we give sum-of-square type loss functions of minimum degree. When \(S\) is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When \(S\) is approximately given by a sample set \(T\), we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.First-order methods for convex optimizationhttps://zbmath.org/1516.900482023-09-19T14:22:37.575876Z"Dvurechensky, Pavel"https://zbmath.org/authors/?q=ai:dvurechensky.pavel-e"Shtern, Shimrit"https://zbmath.org/authors/?q=ai:shtern.shimrit"Staudigl, Mathias"https://zbmath.org/authors/?q=ai:staudigl.mathiasSummary: First-order methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. First-order methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in large-scale optimization problems. In this survey, we cover a number of key developments in gradient-based optimization methods. This includes non-Euclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projection-free methods, and proximal versions of primal-dual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms.Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradienthttps://zbmath.org/1516.900492023-09-19T14:22:37.575876Z"Fercoq, Olivier"https://zbmath.org/authors/?q=ai:fercoq.olivierSummary: We study the linear convergence of the primal-dual hybrid gradient method. After a review of current analyses, we show that they do not explain properly the behavior of the algorithm, even on the most simple problems. We thus introduce the quadratic error bound of the smoothed gap, a new regularity assumption that holds for a wide class of optimization problems. Equipped with this tool, we manage to prove tighter convergence rates. Then, we show that averaging and restarting the primal-dual hybrid gradient allows us to leverage better the regularity constant. Numerical experiments on linear and quadratic programs, ridge regression and image denoising illustrate the findings of the paper.Survey descent: a multipoint generalization of gradient descent for nonsmooth optimizationhttps://zbmath.org/1516.900512023-09-19T14:22:37.575876Z"Han, X. Y."https://zbmath.org/authors/?q=ai:han.xiaoyan"Lewis, Adrian S."https://zbmath.org/authors/?q=ai:lewis.adrian-sSummary: For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable, and the number of cutting planes invoked by traditional remedies is difficult to bound, leading to convergence guarantees that are sublinear relative to the cumulative number of gradient evaluations. We instead propose a multipoint generalization of the gradient descent iteration for local optimization. While our iteration was designed with general objectives in mind, we are motivated by a ``max-of-smooth'' model that captures the subdifferential dimension at optimality. We prove linear convergence when the objective is itself max-of-smooth, and experiments suggest a more general phenomenon.Zeroth-order nonconvex stochastic optimization: handling constraints, high dimensionality, and saddle pointshttps://zbmath.org/1516.900562023-09-19T14:22:37.575876Z"Balasubramanian, Krishnakumar"https://zbmath.org/authors/?q=ai:balasubramanian.krishnakumar"Ghadimi, Saeed"https://zbmath.org/authors/?q=ai:ghadimi.saeedSummary: In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting, and saddle point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To facilitate zeroth-order optimization in high dimensions, we explore the advantages of structural sparsity assumptions. Specifically, (i) we highlight an implicit regularization phenomenon where the standard stochastic gradient algorithm with zeroth-order information adapts to the sparsity of the problem at hand by just varying the step size and (ii) propose a truncated stochastic gradient algorithm with zeroth-order information, whose rate of convergence depends only poly-logarithmically on the dimensionality. We next focus on avoiding saddle points in nonconvex setting. Toward that, we interpret the Gaussian smoothing technique for estimating gradient based on zeroth-order information as an instantiation of first-order Stein's identity. Based on this, we provide a novel linear-(in dimension) time estimator of the Hessian matrix of a function using only zeroth-order information, which is based on second-order Stein's identity. We then provide a zeroth-order variant of cubic regularized Newton method for avoiding saddle points and discuss its rate of convergence to local minima.A bundle trust region algorithm for minimizing locally Lipschitz functionshttps://zbmath.org/1516.900632023-09-19T14:22:37.575876Z"Hoseini Monjezi, Najmeh"https://zbmath.org/authors/?q=ai:hoseini-monjezi.najmehSummary: We propose a new algorithm for minimizing locally Lipschitz functions that combines both the bundle and trust region techniques. Based on the bundle methods the objective function is approximated by a piecewise linear working model which is updated by adding cutting planes at unsuccessful trial steps. The algorithm defines, at each iteration, a new trial point by solving a subproblem that employs the working model in the objective function subject to a region, which is called the trust region. The algorithm is studied from both theoretical and practical points of view. Under a lower-\(C^1\) assumption on the objective function, global convergence of it is verified to stationary points. In order to demonstrate the reliability and efficiency of the proposed algorithm, a MATLAB implementation of it is prepared and numerical experiments have been made using some academic nonsmooth test problems. Computational results show that the developed method is efficient for solving nonsmooth and nonconvex optimization problems.On global convergence of alternating least squares for tensor approximationhttps://zbmath.org/1516.900682023-09-19T14:22:37.575876Z"Yang, Yuning"https://zbmath.org/authors/?q=ai:yang.yuningSummary: Alternating least squares is a classic, easily implemented, yet widely used method for tensor canonical polyadic approximation. Its subsequential and global convergence is ensured if the partial Hessians of the blocks during the whole sequence are uniformly positive definite. This paper shows that this positive definiteness assumption can be weakened in two ways. Firstly, if the smallest positive eigenvalues of the partial Hessians are uniformly positive, and the solutions of the subproblems are properly chosen, then global convergence holds. This allows the partial Hessians to be only positive semidefinite. Next, if at a limit point, the partial Hessians are positive definite, then global convergence also holds. We also discuss the connection of such an assumption to the uniqueness of exact CP decomposition.Inexact gradient projection method with relative error tolerancehttps://zbmath.org/1516.900892023-09-19T14:22:37.575876Z"Aguiar, A. A."https://zbmath.org/authors/?q=ai:aguiar.a-a"Ferreira, O. P."https://zbmath.org/authors/?q=ai:ferreira.orizon-pereira"Prudente, L. F."https://zbmath.org/authors/?q=ai:prudente.leandro-fSummary: A gradient projection method with feasible inexact projections is proposed in the present paper. The inexact projection is performed using a general relative error tolerance. Asymptotic convergence analysis under quasiconvexity assumption and iteration-complexity bounds under convexity assumption of the method employing constant and Armijo step sizes are presented. Numerical results are reported illustrating the potential advantages of considering inexact projections instead of exact ones in some medium scale instances of a least squares problem over the spectrohedron.Strict constraint qualifications and sequential optimality conditions for constrained optimizationhttps://zbmath.org/1516.900912023-09-19T14:22:37.575876Z"Andreani, Roberto"https://zbmath.org/authors/?q=ai:andreani.roberto"Martínez, José Mario"https://zbmath.org/authors/?q=ai:martinez.jose-mario"Ramos, Alberto"https://zbmath.org/authors/?q=ai:ramos.alberto"Silva, Paulo J. S."https://zbmath.org/authors/?q=ai:silva.paulo-j-sFor finite-dimensional equality and inequality constrained optimization problems with continuously differentiable objective and constraint functions, the paper considers strict constraint qualifications, that is, conditions ensuring that a point satisfying a sequential optimality condition is necessarily a KKT point. The main results present the weakest strict constraint qualifications associated with a number of sequential optimality conditions used to establish stopping criteria in numerical algorithms. The authors also study the logical relations existing among the new constraint qualifications as well as those linking them to other constraint qualifications previously considered in the literature.
Reviewer: Juan-Enrique Martínez-Legaz (Barcelona)A new scalarization approach and applications in set optimizationhttps://zbmath.org/1516.900922023-09-19T14:22:37.575876Z"Anh, Lam Quoc"https://zbmath.org/authors/?q=ai:anh.lam-quoc"Duoc, Pham Thanh"https://zbmath.org/authors/?q=ai:duoc.pham-thanh"Duong, Tran Thi Thuy"https://zbmath.org/authors/?q=ai:duong.tran-thi-thuySummary: In this paper, we introduce a new nonlinear scalarization function and apply it to investigate optimality conditions and scalar representations for minimal sets and set optimization problems with the set less order relation. First, we propose a new scalarization function for sets belonging to the power set of a normed space and discuss its properties. Next, we apply such properties to formulate sufficient and necessary optimality conditions for strictly minimal and weakly minimal sets. Finally, under suitable adjustments, we provide nonlinear scalarization functions and investigate scalar representations of strictly efficient solution and weakly efficient solution sets of the reference problems.A unified analysis of descent sequences in weakly convex optimization, including convergence rates for bundle methodshttps://zbmath.org/1516.900942023-09-19T14:22:37.575876Z"Atenas, Felipe"https://zbmath.org/authors/?q=ai:atenas.felipe"Sagastizábal, Claudia"https://zbmath.org/authors/?q=ai:sagastizabal.claudia-a"Silva, Paulo J. S."https://zbmath.org/authors/?q=ai:silva.paulo-j-s"Solodov, Mikhail"https://zbmath.org/authors/?q=ai:solodov.mikhail-vSummary: We present a framework for analyzing convergence and local rates of convergence of a class of descent algorithms, assuming the objective function is weakly convex. The framework is general, in the sense that it combines the possibility of explicit iterations (based on the gradient or a subgradient at the current iterate), implicit iterations (using a subgradient at the next iteration, like in the proximal schemes), as well as iterations when the associated subgradient is specially constructed and does not correspond either to the current or the next point (this is the case of descent steps in bundle methods). Under the subdifferential-based error bound on the distance to critical points, linear rates of convergence are established. Our analysis applies, among other techniques, to prox-descent for decomposable functions, the proximal-gradient method for a sum of functions, redistributed bundle methods, and a class of algorithms that can be cast in the feasible descent framework for constrained optimization.An abstract convergence framework with application to inertial inexact forward-backward methodshttps://zbmath.org/1516.900952023-09-19T14:22:37.575876Z"Bonettini, Silvia"https://zbmath.org/authors/?q=ai:bonettini.silvia"Ochs, Peter"https://zbmath.org/authors/?q=ai:ochs.peter"Prato, Marco"https://zbmath.org/authors/?q=ai:prato.marco"Rebegoldi, Simone"https://zbmath.org/authors/?q=ai:rebegoldi.simoneSummary: In this paper we introduce a novel abstract descent scheme suited for the minimization of proper and lower semicontinuous functions. The proposed abstract scheme generalizes a set of properties that are crucial for the convergence of several first-order methods designed for nonsmooth nonconvex optimization problems. Such properties guarantee the convergence of the full sequence of iterates to a stationary point, if the objective function satisfies the Kurdyka-Łojasiewicz property. The abstract framework allows for the design of new algorithms. We propose two inertial-type algorithms with implementable inexactness criteria for the main iteration update step. The first algorithm, \(\mathrm{i}^2\)Piano, exploits large steps by adjusting a local Lipschitz constant. The second algorithm, iPila, overcomes the main drawback of line-search based methods by enforcing a descent only on a merit function instead of the objective function. Both algorithms have the potential to escape local minimizers (or stationary points) by leveraging the inertial feature. Moreover, they are proved to enjoy the full convergence guarantees of the abstract descent scheme, which is the best we can expect in such a general nonsmooth nonconvex optimization setup using first-order methods. The efficiency of the proposed algorithms is demonstrated on two exemplary image deblurring problems, where we can appreciate the benefits of performing a linesearch along the descent direction inside an inertial scheme.Error bounds for the solution sets of generalized polynomial complementarity problemshttps://zbmath.org/1516.901082023-09-19T14:22:37.575876Z"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie.7Summary: In this paper, several error bounds for the solution sets of the generalized polynomial complementarity problems (GPCPs) with explicit exponents are given. As the solution set of a GPCP is the solution set of a system of polynomial equalities and inequalities, the state-of-art results in error bounds for polynomial systems can be applied directly. Starting from this, a much better error bound result for the solution set of a GPCP based on exploring the intrinsic sparsity via tensor decomposition is established.Convex generalized Nash equilibrium problems and polynomial optimizationhttps://zbmath.org/1516.910072023-09-19T14:22:37.575876Z"Nie, Jiawang"https://zbmath.org/authors/?q=ai:nie.jiawang"Tang, Xindong"https://zbmath.org/authors/?q=ai:tang.xindongSummary: This paper studies convex generalized Nash equilibrium problems that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing generalized Nash equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.Pricing high-dimensional Bermudan options with hierarchical tensor formatshttps://zbmath.org/1516.910662023-09-19T14:22:37.575876Z"Bayer, Christian"https://zbmath.org/authors/?q=ai:bayer.christian"Eigel, Martin"https://zbmath.org/authors/?q=ai:eigel.martin"Sallandt, Leon"https://zbmath.org/authors/?q=ai:sallandt.leon"Trunschke, Philipp"https://zbmath.org/authors/?q=ai:trunschke.philippSummary: An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the ``curse of dimensionality'' can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favorable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent neural network based methods.State-dependent Riccati equation feedback stabilization for nonlinear PDEshttps://zbmath.org/1516.932032023-09-19T14:22:37.575876Z"Alla, Alessandro"https://zbmath.org/authors/?q=ai:alla.alessandro"Kalise, Dante"https://zbmath.org/authors/?q=ai:kalise.dante"Simoncini, Valeria"https://zbmath.org/authors/?q=ai:simoncini.valeriaIn this paper the authors study the synthesis issue of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs. By using algebraic Riccati equation and Lyapunov equation, an approach based on the state-dependent Riccati equation (SDRE) is proposed. Depending on the nonlinearity and the dimension of the resulting problem, offline, online, and hybrid offline-online alternatives to the SDRE synthesis are presented. The hybrid offline-online SDRE method reduces to the sequential solution of Lyapunov equations, effectively enabling the computation of suboptimal feedback controls for two-dimensional PDEs. Numerical tests for the sine-Gordon, degenerate Zeldovich, and viscous Burgers' PDEs are given, in particular, a thorough experimental assessment of the proposed methodology is provided.
Reviewer: Gen Qi Xu (Tianjin)Parameter learning and fractional differential operators: applications in regularized image denoising and decomposition problemshttps://zbmath.org/1516.940032023-09-19T14:22:37.575876Z"Bartels, Sören"https://zbmath.org/authors/?q=ai:bartels.soren"Weber, Nico"https://zbmath.org/authors/?q=ai:weber.nicoSummary: In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model [\textit{L. I. Rudin} et al., Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)]. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.Joint reconstruction-segmentation on graphshttps://zbmath.org/1516.940042023-09-19T14:22:37.575876Z"Budd, Jeremy M."https://zbmath.org/authors/?q=ai:budd.jeremy-m"van Gennip, Yves"https://zbmath.org/authors/?q=ai:van-gennip.yves"Latz, Jonas"https://zbmath.org/authors/?q=ai:latz.jonas"Parisotto, Simone"https://zbmath.org/authors/?q=ai:parisotto.simone"Schönlieb, Carola-Bibiane"https://zbmath.org/authors/?q=ai:schonlieb.carola-bibianeSummary: Practical image segmentation tasks concern images which must be reconstructed from noisy, distorted, and/or incomplete observations. A recent approach for solving such tasks is to perform this reconstruction jointly with the segmentation, using each to guide the other. However, this work has so far employed relatively simple segmentation methods, such as the Chan-Vese algorithm. In this paper, we present a method for joint reconstruction-segmentation using graph-based segmentation methods, which have been seeing increasing recent interest. Complications arise due to the large size of the matrices involved, and we show how these complications can be managed. We then analyze the convergence properties of our scheme. Finally, we apply this scheme to distorted versions of ``two cows'' images familiar from previous graph-based segmentation literature, first to a highly noised version and second to a blurred version, achieving highly accurate segmentations in both cases. We compare these results to those obtained by sequential reconstruction-segmentation approaches, finding that our method competes with, or even outperforms, those approaches in terms of reconstruction and segmentation accuracy.Anisotropic Chan-Vese segmentationhttps://zbmath.org/1516.940052023-09-19T14:22:37.575876Z"Moll, Salvador"https://zbmath.org/authors/?q=ai:moll.salvador"Pallardó-Julià, Vicent"https://zbmath.org/authors/?q=ai:pallardo-julia.vicentSummary: In this paper we study a variant to \textit{Chan-Vese} (CV) segmentation model with rectilinear anisotropy. We show existence of minimizers in the 2-phases case and how they are related to the (anisotropic) Rudin-Osher-Fatemi (ROF) denoising model [\textit{L. I. Rudin} et al., Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)]. Our analysis shows that in the natural case of a piecewise constant on rectangles image \(( \operatorname{PCR}\) function in short), there exists a minimizer of the CV functional which is also piecewise constant on rectangles over the same grid that the one defined by the original image. In the multiphase case, we show that minimizers of the CV multiphase functional also share this property in the case that the initial image is a \(\operatorname{PCR}\) function. We also investigate a multiphase and anisotropic version of the Truncated ROF algorithm, and we compare the solutions given by this algorithm with minimizers of the multiphase anisotropic CV functional.Image scaling by de la Vallée-Poussin filtered interpolationhttps://zbmath.org/1516.940062023-09-19T14:22:37.575876Z"Occorsio, Donatella"https://zbmath.org/authors/?q=ai:occorsio.donatella"Ramella, Giuliana"https://zbmath.org/authors/?q=ai:ramella.giuliana"Themistoclakis, Woula"https://zbmath.org/authors/?q=ai:themistoclakis.woulaSummary: We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. The resized image is achieved by sampling a bivariate polynomial which globally interpolates the data at the new scale. The method's particularities lay in both the sampling model and the interpolation polynomial we use. Rather than classical uniform grids, we consider an unusual sampling system based on Chebyshev zeros of the first kind. Such optimal distribution of nodes permits to consider near-best interpolation polynomials defined by a filter of de la Vallée-Poussin type. The action ray of this filter provides an additional parameter that can be suitably regulated to improve the approximation. The method has been tested on a significant number of different image datasets. The results are evaluated in qualitative and quantitative terms and compared with other available competitive methods. The perceived quality of the resulting scaled images is such that important details are preserved, and the appearance of artifacts is low. Competitive quality measurement values, good visual quality, limited computational effort, and moderate memory demand make the method suitable for real-world applications.Nearly optimal bounds for the global geometric landscape of phase retrievalhttps://zbmath.org/1516.940082023-09-19T14:22:37.575876Z"Cai, Jian-Feng"https://zbmath.org/authors/?q=ai:cai.jian-feng"Huang, Meng"https://zbmath.org/authors/?q=ai:huang.meng"Li, Dong"https://zbmath.org/authors/?q=ai:li.dong"Wang, Yang"https://zbmath.org/authors/?q=ai:wang.yang.1Summary: The phase retrieval problem is concerned with recovering an unknown signal \(\mathfrak{x}\in\mathbb{C}^n\) from a set of magnitude-only measurements \(y_j|\langle\mathfrak{a}_j,\mathfrak{x}\rangle|\), \(j=1,\dots,m\). A natural least squares formulation can be used to solve this problem efficiently even with random initialization, despite its non-convexity of the loss function. One way to explain this surprising phenomenon is the benign geometric landscape: (1) all local minimizers are global; and (2) the objective function has a negative curvature around each saddle point and local maximizer. In this paper, we show that \(m=O(n\log n)\) Gaussian random measurements are sufficient to guarantee the loss function of a commonly used estimator has such benign geometric landscape with high probability. This is a step toward answering the open problem given by \textit{J. Sun} et al. [Found. Comput. Math. 18, No. 5, 1131--1198 (2018; Zbl 1401.94049)], in which the authors suggest that \(O(n\log n)\) or even \(O(n)\) is enough to guarantee the favorable geometric property.Stochastic greedy algorithms for multiple measurement vectorshttps://zbmath.org/1516.940142023-09-19T14:22:37.575876Z"Qin, Jing"https://zbmath.org/authors/?q=ai:qin.jing"Li, Shuang"https://zbmath.org/authors/?q=ai:li.shuang"Needell, Deanna"https://zbmath.org/authors/?q=ai:needell.deanna"Ma, Anna"https://zbmath.org/authors/?q=ai:ma.anna"Grotheer, Rachel"https://zbmath.org/authors/?q=ai:grotheer.rachel"Huang, Chenxi"https://zbmath.org/authors/?q=ai:huang.chenxi"Durgin, Natalie"https://zbmath.org/authors/?q=ai:durgin.natalieSummary: Sparse representation of a single measurement vector (SMV) has been explored in a variety of compressive sensing applications. Recently, SMV models have been extended to solve multiple measurement vectors (MMV) problems, where the underlying signal is assumed to have joint sparse structures. To circumvent the NP-hardness of the \(\ell_0\) minimization problem, many deterministic MMV algorithms solve the convex relaxed models with limited efficiency. In this paper, we develop stochastic greedy algorithms for solving the joint sparse MMV reconstruction problem. In particular, we propose the MMV Stochastic Iterative Hard Thresholding (MStoIHT) and MMV Stochastic Gradient Matching Pursuit (MStoGradMP) algorithms, and we also utilize the mini-batching technique to further improve their performance. Convergence analysis indicates that the proposed algorithms are able to converge faster than their SMV counterparts, i.e., concatenated StoIHT and StoGradMP, under certain conditions. Numerical experiments have illustrated the superior effectiveness of the proposed algorithms over their SMV counterparts.Modeling and simulating the sample complexity of solving LWE using BKW-style algorithmshttps://zbmath.org/1516.940332023-09-19T14:22:37.575876Z"Guo, Qian"https://zbmath.org/authors/?q=ai:guo.qian"Mårtensson, Erik"https://zbmath.org/authors/?q=ai:martensson.erik"Stankovski Wagner, Paul"https://zbmath.org/authors/?q=ai:stankovski-wagner.paulSummary: The Learning with Errors (LWE) problem receives much attention in cryptography, mainly due to its fundamental significance in post-quantum cryptography. Among its solving algorithms, the Blum-Kalai-Wasserman (BKW) algorithm, originally proposed for solving the Learning Parity with Noise (LPN) problem, performs well, especially for certain parameter settings with cryptographic importance. The BKW algorithm consists of two phases, the reduction phase and the solving phase. In this work, we study the performance of distinguishers used in the solving phase. We show that the Fast Fourier Transform (FFT) distinguisher from [\textit{A. Duc} et al., Lect. Notes Comput. Sci. 9056, 173--202 (2015; Zbl 1365.94424)] has the same sample complexity as the optimal distinguisher, when making the same number of hypotheses. We also show via simulation that it performs much better than previous theory predicts and develop a sample complexity model that matches the simulations better. We also introduce an improved, pruned version of the FFT distinguisher. Finally, we indicate, via extensive experiments, that the sample dependency due to both LF2 and sample amplification is limited.