Recent zbMATH articles in MSC 65https://zbmath.org/atom/cc/652022-09-13T20:28:31.338867ZUnknown authorWerkzeugApproximation methods in science and engineeringhttps://zbmath.org/1491.000032022-09-13T20:28:31.338867Z"Jazar, Reza N."https://zbmath.org/authors/?q=ai:jazar.reza-nPublisher's description: \textit{Approximation Methods in Engineering and Science} covers fundamental and advanced topics in three areas: Dimensional Analysis, Continued Fractions, and Stability Analysis of the Mathieu Differential Equation. Throughout the book, a strong emphasis is given to concepts and methods used in everyday calculations. Dimensional analysis is a crucial need for every engineer and scientist to be able to do experiments on scaled models and use the results in real world applications. Knowing that most nonlinear equations have no analytic solution, the power series solution is assumed to be the first approach to derive an approximate solution. However, this book will show the advantages of continued fractions and provides a systematic method to develop better approximate solutions in continued fractions. It also shows the importance of determining stability chart of the Mathieu equation and reviews and compares several approximate methods for that. The book provides the energy-rate method to study the stability of parametric differential equations that generates much better approximate solutions.
\begin{itemize}
\item
Covers practical model-prototype analysis and nondimensionalization of differential equations;
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Coverage includes approximate methods of responses of nonlinear differential equations;
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Discusses how to apply approximation methods to analysis, design, optimization, and control problems;
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Discusses how to implement approximation methods to new aspects of engineering and physics including nonlinear vibration and vehicle dynamics.
\end{itemize}Book review of: A. Kunoth (ed.) et al., Splines and PDEs: from approximation theory to numerical linear algebrahttps://zbmath.org/1491.000122022-09-13T20:28:31.338867Z"Huyer, W."https://zbmath.org/authors/?q=ai:huyer.waltraudReview of [Zbl 1404.65003].Book review of: E. Smith, Introduction to the tools of scientific computinghttps://zbmath.org/1491.000152022-09-13T20:28:31.338867Z"Mang, Andreas"https://zbmath.org/authors/?q=ai:mang.andreasReview of [Zbl 1483.65006].Book review of: S. M. Steward, How to integrate it. A practical guide to finding elementary integralshttps://zbmath.org/1491.000182022-09-13T20:28:31.338867Z"Pineda-Villavicencio, Guillermo"https://zbmath.org/authors/?q=ai:pineda-villavicencio.guillermoReview of [Zbl 1404.26004].Book review of: B. G. Osgood, Lectures on the Fourier transform and its applicationshttps://zbmath.org/1491.000232022-09-13T20:28:31.338867Z"Rindler, H."https://zbmath.org/authors/?q=ai:rindler.haraldReview of [Zbl 1412.42003].Book review of: M. Asadzadeh, An introduction to the finite element method for differential equationshttps://zbmath.org/1491.000262022-09-13T20:28:31.338867Z"Sachs, Ekkehard"https://zbmath.org/authors/?q=ai:sachs.ekkehard-wReview of [Zbl 1446.65001].Book review of: N. Gillis, Nonnegative matrix factorizationhttps://zbmath.org/1491.000272022-09-13T20:28:31.338867Z"Saibara, Arvind K."https://zbmath.org/authors/?q=ai:saibara.arvind-kReview of [Zbl 1470.68009].To the special issue dedicated to the 3rd international conference ``Numerical computations: theory and algorithms -- NUMTA 2019'' June 15--21, 2019, Isola Capo Rizzuto, Italyhttps://zbmath.org/1491.000512022-09-13T20:28:31.338867Z(no abstract)Elementary linear algebrahttps://zbmath.org/1491.150012022-09-13T20:28:31.338867Z"Andrilli, Stephen"https://zbmath.org/authors/?q=ai:andrilli.stephen"Hecker, David"https://zbmath.org/authors/?q=ai:hecker.davidPublisher's description: Elementary Linear Algebra, Sixth Edition provides a solid introduction to both the computational and theoretical aspects of linear algebra, covering many important real-world applications, including graph theory, circuit theory, Markov chains, elementary coding theory, least-squares polynomials and least-squares solutions for inconsistent systems, differential equations, computer graphics and quadratic forms. In addition, many computational techniques in linear algebra are presented, including iterative methods for solving linear systems, LDU Decomposition, the Power Method for finding eigenvalues, QR Decomposition, and Singular Value Decomposition and its usefulness in digital imaging.
See the reviews of the 1st and 2nd editions in [Zbl 0787.15001; Zbl 0963.15001]. For the 4th and 5th editions see [Zbl 1204.15001; Zbl 1338.15001].Methods for solving \textit{LR}-bipolar fuzzy linear systemshttps://zbmath.org/1491.150062022-09-13T20:28:31.338867Z"Akram, Muhammad"https://zbmath.org/authors/?q=ai:akram.muhammad"Allahviranloo, Tofigh"https://zbmath.org/authors/?q=ai:allahviranloo.tofigh"Pedrycz, Witold"https://zbmath.org/authors/?q=ai:pedrycz.witold"Ali, Muhammad"https://zbmath.org/authors/?q=ai:ali.muhammad-irfan|ali.muhammad-syed|ali.muhammad-zubair|ali.muhammad-i|ali.muhammad-asim|ali.muhammad-aamir|ali.muhammad-arfan|ali.muhammad-nasir|ali.muhammad-usmanSummary: In this paper, we propose a technique to solve \textit{LR}-bipolar fuzzy linear system (BFLS), \textit{LR}-complex bipolar fuzzy linear (CBFL) system with real coefficients and \textit{LR}-complex bipolar fuzzy linear (CBFL) system with complex coefficients of equations. Initially, we solve the \textit{LR}-BFLS of equations using a pair of positive \((*)\) and negative \((\bullet )\) of two \(n \times n\) \textit{LR}-real linear systems by using mean values and left-right spread systems. We also provide the necessary and sufficient conditions for the solution of \textit{LR}-BFLS of equations. We illustrate the method by using some numerical examples of symmetric and asymmetric \textit{LR}-BFLS equations and obtain the strong and weak solutions to the systems. Further, we solve the \textit{LR}-CBFL system of equations with real coefficients and \textit{LR}-CBFL system of equations with complex coefficients by pair of positive \((*)\) and negative \((\bullet )\) two \(n \times n\) real and complex \textit{LR}-bipolar fuzzy linear systems by using mean values and left-right spread systems. Finally, we show the usage of technique to solve the current flow circuit which is represented by \textit{LR}-CBFL system with complex coefficients and obtain the unknown current in term of \textit{LR}-bipolar fuzzy complex number.On the WZ factorization of the real and integer matriceshttps://zbmath.org/1491.150072022-09-13T20:28:31.338867Z"Babolian, E."https://zbmath.org/authors/?q=ai:babolian.esmaeil|babolian.esmail"Golpar-Raboky, E."https://zbmath.org/authors/?q=ai:raboky.e-golpar|golpar-raboky.effatSummary: The \textit{QIF} (Quadrant Interlocking Factorization) method of \textit{D. J. Evans} and \textit{M. Hatzopoulos} [Comput. J. 19, 184--187 (1976; Zbl 0322.65015)]
solves linear equation systems using \textit{WZ} factorization. The WZ factorization can be faster than the \textit{LU} factorization because, it performs the simultaneous evaluation of two columns or two rows. Here, we present a method for computing the real and integer \textit{WZ} and \textit{ZW} factorizations by using the null space generators of some special nested submatrices of a matrix \textit{A}.Extensions of generalized core-EP inversehttps://zbmath.org/1491.150102022-09-13T20:28:31.338867Z"Mosić, Dijana"https://zbmath.org/authors/?q=ai:mosic.dijana"Stanimirović, Predrag S."https://zbmath.org/authors/?q=ai:stanimirovic.predrag-s"Zhang, Daochang"https://zbmath.org/authors/?q=ai:zhang.daochangSummary: The intention of our research is to define and investigate some extensions of generalized core-EP (or shortly GCEP) inverse. Inspired by Urquhart expression for the outer inverses, we firstly introduce \(\Phi_1\)-GCEP inverse replacing \(A^{(2)}_{\mathcal{R}(B), \mathcal{N}(C)}\) in the definition of the GCEP inverse by the more general expression \(\Phi_1 := B(CAB)^{(1)}C\). Further, \(\Phi_2\)-GCEP inverse is defined stating \(\Phi_2 := B(CAB)^{(2)}C\) instead of \(A^{(2)}_{\mathcal{R}(B), \mathcal{N}(C)}\) in GCEP inverse. The most comprehensive form of the GCEP inverse is presented as \(\Phi\)-GCEP inverse based on the replacement \(A^{(2)}_{\mathcal{R}(B), \mathcal{N}(C)}\) with arbitrary matrix \(\Phi\) with an adequate size. Main properties, representations and characterizations of \(\Phi_1\)-GCEP inverse, \(\Phi_2\)-GCEP inverse and \(\Phi\)-GCEP inverse are established as well as algorithms for their calculating. Solvability of some linear matrix equations are investigated applying these three new outer inverses.Annulus containing all the eigenvalues of a matrix polynomialhttps://zbmath.org/1491.150142022-09-13T20:28:31.338867Z"Hans, Sunil"https://zbmath.org/authors/?q=ai:hans.sunil"Raouafi, Samir"https://zbmath.org/authors/?q=ai:raouafi.samirSummary: In this paper, we prove a more general result concerning the location of the eigenvalues of a matrix polynomial in an annulus from which we deduce an interesting result due to \textit{N. J. Higham} and \textit{F. Tisseur} [Linear Algebra Appl. 358, No. 1--3, 5--22 (2003; Zbl 1055.15030)]. Several other known results have been extended to matrix polynomials, which in particular include extension and generalization of a classical result of Cauchy. We also present two examples of matrix polynomials to show that the bounds obtained are close to the actual bounds.Correction to: ``On the unitary block-decomposability of 1-parameter matrix flows and static matrices''https://zbmath.org/1491.150182022-09-13T20:28:31.338867Z"Uhlig, Frank"https://zbmath.org/authors/?q=ai:uhlig.frankCorrection to the author's paper [ibid. 89, No. 2, 529--549 (2022; Zbl 1486.15019)].Accelerating Fourier-Motzkin elimination using bit pattern treeshttps://zbmath.org/1491.150252022-09-13T20:28:31.338867Z"Bastrakov, S. I."https://zbmath.org/authors/?q=ai:bastrakov.sergei-ivanovich"Churkin, A. V."https://zbmath.org/authors/?q=ai:churkin.a-v"Zolotykh, N. Yu."https://zbmath.org/authors/?q=ai:zolotykh.nikolai-yurevichSummary: The paper concerns the elimination of a set of variables from a system of linear inequalities. We employ the widely used Fourier-Motzkin elimination method extended with the Chernikov rules. A straightforward implementation of the algorithm results in extensive enumeration during the most computationally demanding stage. We propose a new way of checking Chernikov rules using bit pattern trees as an accelerating data structure to avoid extensive enumeration. The bit pattern tree is a data structure based on \(k\)-d tree used to accelerate the double description method. First we describe an adaptation of that approach to check the second Chernikov rule in Fourier-Motzkin elimination. We also propose a new algorithm that employs bit pattern trees to accelerate both Chernikov rules. Presented results of computational evaluation prove competitiveness of the proposed algorithms.On variation of eigenvalues of birth and death matrices and random walk matriceshttps://zbmath.org/1491.150382022-09-13T20:28:31.338867Z"Castillo, K."https://zbmath.org/authors/?q=ai:castillo.kenier"Zaballa, I."https://zbmath.org/authors/?q=ai:zaballa.ionSummary: The purpose of this note is twofold: firstly to improve the known results on variation of extreme eigenvalues of birth and death matrices and random walk matrices; and secondly to progress towards the solution of a thirty years old open problem concerning the variation of eigenvalues of these matrices.On a discrete composition of the fractional integral and Caputo derivativehttps://zbmath.org/1491.260062022-09-13T20:28:31.338867Z"Płociniczak, Łukasz"https://zbmath.org/authors/?q=ai:plociniczak.lukaszSummary: We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler-Maclaurin summation formula.Mathematical modelling with differential equationshttps://zbmath.org/1491.340022022-09-13T20:28:31.338867Z"Mickens, Ronald E."https://zbmath.org/authors/?q=ai:mickens.ronald-ePublisher's description: Mathematical Modelling with Differential Equations aims to introduce various strategies for modelling systems using differential equations. Some of these methodologies are elementary and quite direct to comprehend and apply while others are complex in nature and require thoughtful, deep contemplation. Many topics discussed in the chapter do not appear in any of the standard textbooks and this provides users an opportunity to consider a more general set of interesting systems that can be modelled. For example, the book investigates the evolution of a ``toy universe,'' discusses why ``alternate futures'' exists in classical physics, constructs approximate solutions to the famous Thomas-Fermi equation using only algebra and elementary calculus, and examines the importance of ``truly nonlinear'' and oscillating systems.
Features
\begin{itemize}
\item Introduces, defines, and illustrates the concept of ``dynamic consistency'' as the foundation of modelling.
\item Can be used as the basis of an upper-level undergraduate course on general procedures for mathematical modelling using differential equations.
\item Discusses the issue of dimensional analysis and continually demonstrates its value for both the construction and analysis of mathematical modelling.
\end{itemize}Dynamic analysis of a fractional order Rössler systemhttps://zbmath.org/1491.340112022-09-13T20:28:31.338867Z"Cheng, Zhixin"https://zbmath.org/authors/?q=ai:cheng.zhixin(no abstract)Boundary value problem: weak solutions induced by fuzzy partitionshttps://zbmath.org/1491.340342022-09-13T20:28:31.338867Z"Nguyen, Linh"https://zbmath.org/authors/?q=ai:nguyen-viet-linh.|nguyen.linh-thi-hoai|nguyen.linh-h|nguyen.linh-tuan|nguyen.linh-trung|nguyen.linh-ngoc|nguyen.linh-viet|nguyen.linh-anh"Perfilieva, Irina"https://zbmath.org/authors/?q=ai:perfilieva.irina-g"Holčapek, Michal"https://zbmath.org/authors/?q=ai:holcapek.michalSummary: The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called ``two dimensional'' approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.A computational study of transmission dynamics for dengue fever with a fractional approachhttps://zbmath.org/1491.340612022-09-13T20:28:31.338867Z"Kumar, Sunil"https://zbmath.org/authors/?q=ai:kumar.sunil"Chauhan, R. P."https://zbmath.org/authors/?q=ai:chauhan.r-p"Singh, Jagdev"https://zbmath.org/authors/?q=ai:singh.jagdev"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.3Summary: Fractional derivatives are considered an influential weapon in terms of analysis of infectious diseases because of their nonlocal nature. The inclusion of the memory effect is the prime advantage of fractional-order derivatives. The main objective of this article is to investigate the transmission dynamics of dengue fever, we consider generalized Caputo-type fractional derivative (GCFD) \((^C\mathrm{D}_0^{\beta,\sigma})\) for alternate representation of dengue fever disease model. We discuss the existence and uniqueness of the solution of model by using fixed point theory. Further, an adaptive predictor-corrector technique is utilized to evaluate the considered model numerically.An SEIR epidemic model of fractional order to analyze the evolution of the Covid-19 epidemic in Argentinahttps://zbmath.org/1491.340642022-09-13T20:28:31.338867Z"Santos, Juan E."https://zbmath.org/authors/?q=ai:santos.juan-enrique"Carcione, José M."https://zbmath.org/authors/?q=ai:carcione.jose-m"Savioli, Gabriela B."https://zbmath.org/authors/?q=ai:savioli.gabriela-b"Gauzellino, Patricia M."https://zbmath.org/authors/?q=ai:gauzellino.patricia-mercedesSummary: A pandemic caused by a new coronavirus (Covid-19) has spread worldwide, inducing an epidemic still active in Argentina. In this chapter, we present a case study using an SEIR (Susceptible-Exposed-Infected-Recovered) diffusion model of fractional order in time to analyze the evolution of the epidemic in Buenos Aires and neighboring areas (Región Metropolitana de Buenos Aires, (RMBA)) comprising about 15 million inhabitants. In the SEIR model, individuals are divided into four classes, namely, susceptible (S), exposed (E), infected (I) and recovered (R). The SEIR model of fractional order allows for the incorporation of memory, with hereditary properties of the system, being a generalization of the classic SEIR first-order system, where such effects are ignored. Furthermore, the fractional model provides one additional parameter to obtain a better fit of the data. The parameters of the model are calibrated by using as data the number of casualties officially reported. Since infinite solutions honour the data, we show a set of cases with different values of the lockdown parameters, fatality rate, and incubation and infectious periods. The different reproduction ratios \(R_0\) and infection fatality rates (IFR) so obtained indicate the results may differ from recent reported values, constituting possible alternative solutions. A comparison with results obtained with the classic SEIR model is also included. The analysis allows us to study how isolation and social distancing measures affect the time evolution of the epidemic.
For the entire collection see [Zbl 1470.92006].Time-harmonic acoustic scattering from locally perturbed half-planeshttps://zbmath.org/1491.351302022-09-13T20:28:31.338867Z"Bao, Gang"https://zbmath.org/authors/?q=ai:bao.gang"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Yin, Tao"https://zbmath.org/authors/?q=ai:yin.taoAn efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equationhttps://zbmath.org/1491.351312022-09-13T20:28:31.338867Z"Berdawood, Karzan"https://zbmath.org/authors/?q=ai:berdawood.karzan"Nachaoui, Abdeljalil"https://zbmath.org/authors/?q=ai:nachaoui.abdeljalil"Saeed, Rostam"https://zbmath.org/authors/?q=ai:saeed.rostam-k"Nachaoui, Mourad"https://zbmath.org/authors/?q=ai:nachaoui.mourad"Aboud, Fatima"https://zbmath.org/authors/?q=ai:aboud.fatimaSummary: Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [\textit{V. A. Kozlov} et al., Comput. Math. Math. Phys. 31, No. 1, 45--2 (1991); translation from Zh. Vychisl. Mat. Mat. Fiz. 31, No. 1, 64--74 (1991; Zbl 0774.65069)]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number \(k\), when the convergence of KMF algorithm's [loc. cit.] is ensured, our algorithm can be used as an acceleration of convergence.
In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.
We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.Directional \(\mathcal{H}^2\) Compression algorithm: optimisations and application to a discontinuous Galerkin BEM for the Helmholtz equationhttps://zbmath.org/1491.351442022-09-13T20:28:31.338867Z"Messaï, Nadir-Alexandre"https://zbmath.org/authors/?q=ai:messai.nadir-alexandre"Pernet, Sebastien"https://zbmath.org/authors/?q=ai:pernet.sebastien"Bouguerra, Abdesselam"https://zbmath.org/authors/?q=ai:bouguerra.abdesselamSummary: This study aimed to specialise a directional \(\mathcal{H}^2(\mathcal{DH}^2)\) compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal \(\mathcal{DH}^2\) approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a \(\mathcal{DH}^2\) gives better computational efficiency than a classical BEM in the case of high-order finite elements and \(hp\) heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splineshttps://zbmath.org/1491.351452022-09-13T20:28:31.338867Z"Nanfuka, Mary"https://zbmath.org/authors/?q=ai:nanfuka.mary"Berntsson, Fredrik"https://zbmath.org/authors/?q=ai:berntsson.fredrik"Mango, John"https://zbmath.org/authors/?q=ai:mango.john-magero(no abstract)Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalueshttps://zbmath.org/1491.351642022-09-13T20:28:31.338867Z"Harris, Isaac"https://zbmath.org/authors/?q=ai:harris.isaacSummary: In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl's law for the Dirichlet eigenvalues. Numerical examples for computing the simply supported vibrating plate eigenvalues for the unit disk and square are presented. In order to test the accuracy of the approximation, we compare the spectral-Galerkin method to the separation of variables for the unit disk. Whereas for the unit square we will numerically test the convergence rate for a variable coefficient problem.Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equationhttps://zbmath.org/1491.351962022-09-13T20:28:31.338867Z"Allendes, Alejandro"https://zbmath.org/authors/?q=ai:allendes.alejandro"Fuica, Francisco"https://zbmath.org/authors/?q=ai:fuica.francisco"Otárola, Enrique"https://zbmath.org/authors/?q=ai:otarola.enriqueAn integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensionshttps://zbmath.org/1491.352632022-09-13T20:28:31.338867Z"Watschinger, Raphael"https://zbmath.org/authors/?q=ai:watschinger.raphael"Of, Günther"https://zbmath.org/authors/?q=ai:of.guntherSummary: While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps, we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.On a semilinear parabolic problem with non-local (Bitsadze-Samarskii type) boundary conditions in more dimensionshttps://zbmath.org/1491.352752022-09-13T20:28:31.338867Z"Slodička, Marián"https://zbmath.org/authors/?q=ai:slodicka.marianSummary: This paper studies a semilinear parabolic equation in a bounded domain \(\Omega\subset\mathbb{R}^d\) along with nonlocal boundary conditions. The boundary values are linked to the values of a solution on an interior \((d-1)\)-dimensional manifold lying inside \(\Omega\). Firstly, the solvability of a steady-state problem is addressed. Secondly, involving the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem in a weighted Hilbert space is shown. Convergence of approximations is addressed and the error estimated are derived. Numerical experiments support the theoretical algorithms.On numerical approximations to fluid-structure interactions involving compressible fluidshttps://zbmath.org/1491.353192022-09-13T20:28:31.338867Z"Schwarzacher, Sebastian"https://zbmath.org/authors/?q=ai:schwarzacher.sebastian"She, Bangwei"https://zbmath.org/authors/?q=ai:she.bangweiSummary: In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions. A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problemhttps://zbmath.org/1491.353402022-09-13T20:28:31.338867Z"Lepe, Felipe"https://zbmath.org/authors/?q=ai:lepe.felipe"Rivera, Gonzalo"https://zbmath.org/authors/?q=ai:rivera.gonzalo"Vellojin, Jesus"https://zbmath.org/authors/?q=ai:vellojin.jesusInexact GMRES iterations and relaxation strategies with fast-multipole boundary element methodhttps://zbmath.org/1491.353502022-09-13T20:28:31.338867Z"Wang, Tingyu"https://zbmath.org/authors/?q=ai:wang.tingyu"Layton, Simon K."https://zbmath.org/authors/?q=ai:layton.simon-k"Barba, Lorena A."https://zbmath.org/authors/?q=ai:barba.lorena-aSummary: Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, \(p\). We take advantage of a unique property of Krylov iterations that allows lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing \(p\). In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between \(1.5 \times\) and \(2.3 \times\) for Laplace problems and between \(2.7 \times\) and \(3.3 \times\) for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, \url{https://github.com/barbagroup/fmm-bem-relaxed}, and we share reproducibility packages for all results in \url{https://github.com/barbagroup/inexact-gmres/}.On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobilityhttps://zbmath.org/1491.353832022-09-13T20:28:31.338867Z"Fagioli, Simone"https://zbmath.org/authors/?q=ai:fagioli.simone"Tse, Oliver"https://zbmath.org/authors/?q=ai:tse.oliverSummary: We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. The solution is obtained as the limit of approximations constructed via a deterministic system of interacting particles that exhibits a gradient flow structure. At the same time, we expose a rigorous gradient flow structure for this class of equations in terms of an Energy-Dissipation balance, which we obtain via the asymptotic convergence of functionals.Transport equations with inflow boundary conditionshttps://zbmath.org/1491.353862022-09-13T20:28:31.338867Z"Scott, L. Ridgway"https://zbmath.org/authors/?q=ai:scott.larkin-ridgway"Pollock, Sara"https://zbmath.org/authors/?q=ai:pollock.saraSummary: We provide bounds in a Sobolev-space framework for transport equations with nontrivial inflow and outflow. We give, for the first time, bounds on the gradient of the solution with the type of inflow boundary conditions that occur in Poiseuille flow. Following ground-breaking work of the late \textit{C. J. Amick} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 473--513 (1977; Zbl 0367.76027)], we name a generalization of this type of flow domain in his honor. We prove gradient bounds in Lebesgue spaces for general Amick domains which are crucial for proving well posedness of the grade-two fluid model. We include a complete review of transport equations with inflow boundary conditions, providing novel proofs in most cases. To illustrate the theory, we review and extend an example of Bernard that clarifies the singularities of solutions of transport equations with nonzero inflow boundary conditions.Monotone decomposition of the Cauchy problem for a hyperbolic equation based on transport equationshttps://zbmath.org/1491.353872022-09-13T20:28:31.338867Z"Shishkin, G. I."https://zbmath.org/authors/?q=ai:shishkin.grigorii-i"Shishkina, L. P."https://zbmath.org/authors/?q=ai:shishkina.lidia-pSummary: For the Cauchy problem for a hyperbolic equation, a multiplicative approach is developed: a monotone decomposition of the problem is constructed since the hyperbolic operator can be represented by a product of transport operators. The problem for the hyperbolic equation is reduced to a system of problems for transport equations -- transport in the direction of the axis \(x\) and transport in the opposite direction of the axis \(x\). Conditions for the monotonicity of each problem for the transport equations and for the entire multiplicative problem are found. Such a decomposition of the Cauchy problem based on transport problems solved one after the other significantly simplifies the solution of the hyperbolic equation, and the problems for the transport equations are monotone thus ensuring the monotonicity of the decomposition of the Cauchy problem for the hyperbolic equation.An orthogonalization-free parallelizable framework for all-electron calculations in density functional theoryhttps://zbmath.org/1491.354022022-09-13T20:28:31.338867Z"Gao, Bin"https://zbmath.org/authors/?q=ai:gao.bin"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Kuang, Yang"https://zbmath.org/authors/?q=ai:kuang.yang"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.1Time-dependent electromagnetic scattering from thin layershttps://zbmath.org/1491.354082022-09-13T20:28:31.338867Z"Nick, Jörg"https://zbmath.org/authors/?q=ai:nick.jorg"Kovács, Balázs"https://zbmath.org/authors/?q=ai:kovacs.balazs"Lubich, Christian"https://zbmath.org/authors/?q=ai:lubich.christianSummary: The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell's equations. The time-dependent boundary integral equation is discretized with Runge-Kutta based convolution quadrature in time and Raviart-Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge-Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.Coupled domain-boundary variational formulations for Hodge-Helmholtz operatorshttps://zbmath.org/1491.354092022-09-13T20:28:31.338867Z"Schulz, Erick"https://zbmath.org/authors/?q=ai:schulz.erick"Hiptmair, Ralf"https://zbmath.org/authors/?q=ai:hiptmair.ralfSummary: We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a ``rough'' surface. The low-frequency robustness of the potential formulation of Maxwell's equations makes this model a promising starting point for Galerkin discretization.Simultaneous recovery of Robin boundary and coefficient for the Laplace equation by shape derivativehttps://zbmath.org/1491.354522022-09-13T20:28:31.338867Z"Fang, Weifu"https://zbmath.org/authors/?q=ai:fang.weifuSummary: We study the simultaneous recovery of the boundary and coefficient of the Robin boundary condition for the Laplace equation from a pair of solution measurements on another part of the boundary. We derive the variational derivatives of the data-fitting objective functional with respect to the Robin boundary and coefficient, which are then used to device a nonlinear conjugate gradient iterative scheme for the numerical recovery of both the Robin boundary and coefficient together. Numerical examples are presented to illustrate the effectiveness of the recovery algorithms.Series reversion in Calderón's problemhttps://zbmath.org/1491.354562022-09-13T20:28:31.338867Z"Garde, Henrik"https://zbmath.org/authors/?q=ai:garde.henrik"Hyvönen, Nuutti"https://zbmath.org/authors/?q=ai:hyvonen.nuuttiSummary: This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is \(\nabla \cdot (A\nabla u)=0\) in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends \(A\) to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inverse problem with increasing accuracy. The convergence of these methods is shown under conditions that ensure the invertibility of the Fréchet derivative of the forward map. The introduced numerical methods are of the same computational complexity as solving the linearised inverse problem. The analogous results are also presented for the smoothened complete electrode model.Analysis of the inverse Born series: an approach through geometric function theoryhttps://zbmath.org/1491.354592022-09-13T20:28:31.338867Z"Hoskins, Jeremy G."https://zbmath.org/authors/?q=ai:hoskins.jeremy-g"Schotland, John C."https://zbmath.org/authors/?q=ai:schotland.john-cDetermination of the initial density in nonlocal diffusion from final time measurementshttps://zbmath.org/1491.354602022-09-13T20:28:31.338867Z"Hrizi, Mourad"https://zbmath.org/authors/?q=ai:hrizi.mourad"Bensalah, Mohamed"https://zbmath.org/authors/?q=ai:bensalah.mohamed-oudi"Hassine, Maatoug"https://zbmath.org/authors/?q=ai:hassine.maatougSummary: This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systemshttps://zbmath.org/1491.370302022-09-13T20:28:31.338867Z"Goufo, Emile Franc Doungmo"https://zbmath.org/authors/?q=ai:doungmo-goufo.emile-francSummary: Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order \(\alpha = 1)\) and this chaos remains in the fractional case \((\alpha = 0.9)\). Moreover the dynamics of the system change depending on the parameter \(\alpha\), leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.Order and chaos in time periodic Hamiltonian systemshttps://zbmath.org/1491.370582022-09-13T20:28:31.338867Z"Tzemos, A. C."https://zbmath.org/authors/?q=ai:tzemos.athanasios-c"Contopoulos, G."https://zbmath.org/authors/?q=ai:contopoulos.georgeSummary: We describe an algorithm that constructs formal (approximate) integrals of motion in time periodic Hamiltonian systems and apply it to a perturbed 1-d harmonic oscillator of frequency \(\omega_1\) of the form \(H = \frac{1}{2}\left(\omega_1^2x^2 + y^2\right) + \varepsilon H_1\), where \(H_1\) is an external time periodic field of frequency \(\omega\) of the form: (1) \(H_1 = -x^4 \cos(\omega t)\), (2) \(H_1 = x^4 \exp\left(\frac{-x^2}{2}\right)\cos(\omega t)\) and (3) truncated Taylor expansions of the exponential of case (2) in powers of \(x\). The formal integrals are given as series in powers of the perturbation parameter \(\varepsilon\), which are constructed to high orders with the computer algebra system Maple. By use of stroboscopic Poincaré sections we find that in all cases ordered and chaotic orbits coexist. The latter are around unstable periodic orbits. Furthermore we find that the orbits close to the origin \((x, y) = (0, 0)\) are ordered and can be represented accurately by invariant curves given by the approximate integral of motion. While in the case (2) the orbits starting far from the origin are bounded and ordered, for every truncated expansion of the exponential these orbits are chaotic for a finite time and then escape to infinity. The various types of orbits are described in detail.Alternative approaches of evaluating the \(0-1\) test for chaoshttps://zbmath.org/1491.370692022-09-13T20:28:31.338867Z"Martinovič, T."https://zbmath.org/authors/?q=ai:martinovic.tomas|martynovych.tSummary: The \(0-1\) test for chaos is increasingly used in applications where it is important to distinguish between chaotic and regular dynamics of the deterministic dynamical system. This test consists of translating the time series into \((p,q)\) plane. Then the boundedness of the object created in the \((p,q)\) plane is inspected by computing mean square displacement and its growth rate. The computation of the mean square displacement for various lag has high computational complexity. In this paper, the inspection of the boundedness using bounding box and centre of gravity is proposed. This method has linear computational complexity and provides more control over the precision of the test through multiple input parameters.Revisiting the relation between the Lyapunov time and the instability timehttps://zbmath.org/1491.370702022-09-13T20:28:31.338867Z"Cincotta, Pablo M."https://zbmath.org/authors/?q=ai:cincotta.pablo-m"Giordano, Claudia M."https://zbmath.org/authors/?q=ai:giordano.claudia-m"Shevchenko, Ivan I."https://zbmath.org/authors/?q=ai:shevchenko.ivan-iSummary: In this effort we focus on the so-called \(T_L\)-\(T_{\mathrm{inst}}\) relationship i.e., any relation among the Lyapunov time and a characteristic instability time of a given dynamical system. By means of extensive numerical simulations with a high-dimensional dynamical system, a 4D symplectic map, we investigate a possible correlation between both time-scales. Herein the instability time is the one associated to diffusion along the homoclinic tangle of the resonances of the system. We found that different laws could fit the computed values, depending mostly on the dynamics of the system when varying the involved parameters; in some small domain of the parameter space a power law appears while in a larger one an exponential relation fits quite well the computed values of \(T_L\) and \(T_{\mathrm{inst}}\). We compare the obtained functional forms of the relationships with those known for lower-dimensional systems and identify typical functional dependences, confirmed analytically.Error analysis of forced discrete mechanical systemshttps://zbmath.org/1491.370712022-09-13T20:28:31.338867Z"Fernández, Javier"https://zbmath.org/authors/?q=ai:fernandez.javier-alejandro"Zurita, Sebastián Elías Graiff"https://zbmath.org/authors/?q=ai:graiff-zurita.sebastian-elias"Grillo, Sergio"https://zbmath.org/authors/?q=ai:grillo.sergio-danielSummary: The purpose of this paper is to perform an error analysis of the variational integrators of mechanical systems subject to external forcing. Essentially, we prove that when a discretization of contact order \(r\) of the Lagrangian and force are used, the integrator has the same contact order. Our analysis is performed first for discrete forced mechanical systems defined over \(TQ\), where we study the existence of flows, the construction and properties of discrete exact systems and the contact order of the flows (variational integrators) in terms of the contact order of the original systems. Then we use those results to derive the corresponding analysis for the analogous forced systems defined over \(Q\times Q\).Deep learning of conjugate mappingshttps://zbmath.org/1491.370742022-09-13T20:28:31.338867Z"Bramburger, Jason J."https://zbmath.org/authors/?q=ai:bramburger.jason-j"Brunton, Steven L."https://zbmath.org/authors/?q=ai:brunton.steven-l"Nathan Kutz, J."https://zbmath.org/authors/?q=ai:kutz.j-nathanSummary: Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincaré mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto-Sivashinsky equation.Estimates for solutions of bi-infinite systems of linear equationshttps://zbmath.org/1491.390012022-09-13T20:28:31.338867Z"Volkov, Yuriy S."https://zbmath.org/authors/?q=ai:volkov.yuri-s|volkov.yu-s"Novikov, Sergey I."https://zbmath.org/authors/?q=ai:novikov.sergey-iSummary: We consider the problem of estimating in uniform norm solutions of nonhomogeneous difference equations. Difference equations are considered as bi-infinite systems of linear algebraic equations. We give the estimate for Laurent's matrices with diagonal dominance. Based on this result and an idea of decomposition of the matrix into a product of matrices associated with factorization of the characteristic polynomial, we propose estimates for any nonsingular banded Laurent matrix. The established estimates are attainable.Rational interpolation operator with finite Lebesgue constanthttps://zbmath.org/1491.410022022-09-13T20:28:31.338867Z"Zhang, Ren-Jiang"https://zbmath.org/authors/?q=ai:zhang.renjiang"Liu, Xing"https://zbmath.org/authors/?q=ai:liu.xingAuthors' abstract: ``Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for
approximating any continuous function, and the convergence rate of the interpolants
approximating a function is obtained using the modulus of continuity. ``
The improved rational interpolant of a function \(f\) considered by the authors, is of the form
\[ T_nf(x)=\sum_{j=0}^n \sigma_j(x) f(x_j),\]
where \(\sigma_j(x_i)=\delta_{i,j}, \;\; i,j=0,...,n\), while for \(x\ne x_i\)
\[\displaystyle \sigma_j(x)=\frac{(-1)^j}{(x-x_j)|x-x_j|}\sum_{i=0}^n \frac{(-1)^i}{ (x-x_i)|x-x_i|}\,.\]
The main result is Thoerem 1. For \(x_i\) evenly spaced, the Lebesgue constant \(\|T_n\|\) is bounded as follows
\[ \frac{21}{ 40} < \|T_n\| < \frac{17}{10},\;\; n \ge 10 \,. \]
Reviewer: Stefano De Marchi (Padova)The essence of invertible frame multipliers in scalabilityhttps://zbmath.org/1491.420442022-09-13T20:28:31.338867Z"Javanshiri, Hossein"https://zbmath.org/authors/?q=ai:javanshiri.hossein"Abolghasemi, Mohammad"https://zbmath.org/authors/?q=ai:abolghasemi.mohammad"Arefijamaal, Ali Akbar"https://zbmath.org/authors/?q=ai:arefijamaal.ali-akbarA separable Hilbert space is denoted by \(\mathcal{H}.\) A Bessel operator is an operator of the form \[ M_{m, \phi, \psi} (f) = \sum_{i = 1}^{\infty} m_i \langle f, \psi_i \rangle \phi_i, \quad f \in \mathcal{H}, \] where \(\phi\) and \(\psi\) are Bessel sequences in \(\mathcal{H}\), and \(m\) is a bounded complex scalar sequence in \(\mathbb{C}\). The first result (Proposition 1) in this paper pertains to the invertibility of a Bessel operator associated with a given sequence of nonzero scalars \(m\) and a frame \(\Phi = \{\phi_n\}\) of \(\mathcal{H}\). Proposition 1 is then applied to obtain several results on scalability of frames. Scalability of frames addresses the question of when, for a given frame \(\Phi\), there exist scalars \(\{c_n\}\), \(c_n \geq 0\), such that \(\{c_n \phi_n \}\) is a tight frame. If \(c_n > 0\) for all \(n\), then \(\Phi\) is called positively scalable, and if \(\textrm{inf}_n c_n > 0\) then \(\Phi\) is called strictly scalable.
\par The authors have shown that positive and strict scalability coincide for all frames \(\{ \phi_n \}_n\) with \(\lim \textrm{inf}_n \| \phi_n \| > 0\) which, in particular, provides some equivalent conditions for positive scalability of certain frames. In the process, the authors were able to completely characterize the scalability of Riesz bases and Riesz frames. Further, in the context of scalability of frames, by using Feichtinger Conjecture, the authors give \(\alpha\) and \(\beta\) such that the elements of the scaling sequence \(\{c_n\}\) should be chosen from the interval \([\alpha, \beta]\) for all but finitely many \(n.\) In their result, the values of \(\alpha\) and \(\beta\) depend on the optimal frame bounds of the frame \(\Phi\). The last result in the paper is of independent interest and gives an explicit algorithm to construct desired invertible multipliers from the given one.
Reviewer: Somantika Datta (Moscow)Approximate solutions of aggregation and breakage population balance equationshttps://zbmath.org/1491.450142022-09-13T20:28:31.338867Z"Kaur, Gurmeet"https://zbmath.org/authors/?q=ai:kaur.gurmeet"Singh, Randhir"https://zbmath.org/authors/?q=ai:singh.randhir"Briesen, Heiko"https://zbmath.org/authors/?q=ai:briesen.heikoSummary: The work aims to present a semi-analytical approach based on the homotopy analysis method for finding closed-form solutions and approximate solutions to aggregation and breakage population balance equations. The population balance equations are specific integro-partial differential equations. In this work, we first transform both the aggregation and breakage models into integral equations. Then the resulting integral equations are solved by the homotopy analysis method (HAM) to get the series solution, which in particular cases eventually converges to the exact solution. The semi-analytical solutions for various benchmark aggregation kernels like the Ruckenstein/Pulvermacher kernel and breakage kernels such as the Austin kernel are provided using HAM.Multiperiodic solutions of quasilinear systems of integro-differential equations with \(D_c\)-operator and \(\epsilon \)-period of hereditarityhttps://zbmath.org/1491.450152022-09-13T20:28:31.338867Z"Sartabanov, Zhaishylyk Almaganbetovich"https://zbmath.org/authors/?q=ai:sartabanov.zh-a"Aitenova, Gulsezim Muratovna"https://zbmath.org/authors/?q=ai:aitenova.gulsezim-muratovna"Abdikalikova, Galiya Amirgalievna"https://zbmath.org/authors/?q=ai:abdikalikova.galiya-amirgalievnaSummary: We investigate a quasilinear system of partial integro-differential equations with the operator of differentiation in the direction of a vector field, which describes the process of hereditary propagation with an \(\epsilon \)-period of heredity. Under some conditions on the input data, conditions for the solvability of the initial problem for a quasilinear system of integro-differential equations are obtained. On this basis, sufficient conditions for the existence of multiperiodic solutions of integro-differential systems are found under the exponential dichotomy additional assumption on the corresponding homogeneous integro-differential system. The unique solvability of an operator equation in the space of smooth multiperiodic functions is proved, to which the main question under consideration reduces. Thus, sufficient conditions are established for the existence of a unique multiperiodical in all time variables solution of a quasilinear system of integro-differential equations with the differentiation operator in the directions of a vector field and a finite period of hereditarity.Method of orthogonal polynomials for an approximate solution of singular integro-differential equations as applied to two-dimensional diffraction problemshttps://zbmath.org/1491.450162022-09-13T20:28:31.338867Z"Rasol'ko, G. A."https://zbmath.org/authors/?q=ai:rasolko.galina-alekseevna"Volkov, V. M."https://zbmath.org/authors/?q=ai:volkov.vasilii-mikhailovichSummary: We consider a mathematical model of scattering of \(H \)-polarized electromagnetic waves by a screen with a curvilinear boundary based on a singular integro-differential equation with a Cauchy kernel and a logarithmic singularity. The integrands contain both the unknown function and its first derivative. For the numerical analysis of this model, two computational schemes are constructed based on the representation of the unknown function in the form of a linear combination of orthogonal Chebyshev polynomials and spectral relations, which permit one to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the basis of Chebyshev polynomials are calculated as a solution of the corresponding system of linear algebraic equations. The results of numerical experiments show that the error in the approximate solution on a grid of 20--30 nodes does not exceed the roundoff error.Graph convergence and generalized Yosida approximation operator with an applicationhttps://zbmath.org/1491.470552022-09-13T20:28:31.338867Z"Ahmad, Rais"https://zbmath.org/authors/?q=ai:ahmad.rais"Ishtyak, Mohd."https://zbmath.org/authors/?q=ai:ishtyak.mohd"Rahaman, Mijanur"https://zbmath.org/authors/?q=ai:rahaman.mijanur"Ahmad, Iqbal"https://zbmath.org/authors/?q=ai:ahmad.iqbalSummary: In this paper, we introduce a Yosida inclusion problem as well as a generalized Yosida approximation operator. Using the graph convergence of \(H(\cdot, \cdot)\)-accretive operator and resolvent operator convergence discussed in [\textit{X. Li} and \textit{N.-j. Huang}, Appl. Math. Comput. 217, No. 22, 9053--9061 (2011; Zbl 1308.47072)], we establish the convergence for generalized Yosida approximation operator. As an application, we solve a Yosida inclusion problem in \(q\)-uniformly smooth Banach spaces. An example is constructed, and through \texttt{MATLAB} programming, we show some graphics for the convergence of generalized Yosida approximation operator.A new gradient projection algorithm for convex minimization problem and its application to split feasibility problemhttps://zbmath.org/1491.470602022-09-13T20:28:31.338867Z"Ertürk, Müzeyyen"https://zbmath.org/authors/?q=ai:erturk.muzeyyen"Kızmaz, Asiye"https://zbmath.org/authors/?q=ai:kizmaz.asiyeSummary: In this paper, we study convergence analysis of a new gradient projection algorithm for solving convex minimization problems in Hilbert spaces. We observe that the proposed gradient projection algorithm weakly converges to a minimum of convex function \(f\) which is defined from a closed and convex subset of a Hilbert space to \(\mathbb{R} \). Also, we give a nontrivial example to illustrate our result in an infinite dimensional Hilbert space. We apply our result to solve the split feasibility problem.Extragradient methods for solving equilibrium problems, variational inequalities, and fixed point problemshttps://zbmath.org/1491.470642022-09-13T20:28:31.338867Z"Jouymandi, Zeynab"https://zbmath.org/authors/?q=ai:jouymandi.zeynab"Moradlou, Fridoun"https://zbmath.org/authors/?q=ai:moradlou.fridounSummary: In this paper, we propose the new extragradient algorithms for an \(\alpha\)-inverse-strongly monotone operator and a relatively nonexpansive mapping in Banach spaces. We prove convergence theorems by this methods under suitable conditions. Applying our algorithms, we find a zero point of maximal monotone operators. Using \texttt{FMINCON} optimization toolbox in \texttt{MATLAB}, we give an example to illustrate the usability of our results.A splitting algorithm for equilibrium problem given by the difference of two bifunctionshttps://zbmath.org/1491.470692022-09-13T20:28:31.338867Z"Pham Ky Anh"https://zbmath.org/authors/?q=ai:pham-ky-anh."Trinh Ngoc Hai"https://zbmath.org/authors/?q=ai:trinh-ngoc-hai.Summary: In this paper, we introduce a splitting algorithm for solving equilibrium problems given by the difference of two bifunctions in a real Hilbert space. Under suitable assumptions on component bifunctions, we prove strong convergence of the proposed algorithm. In contrast to most existing projection-type methods for equilibrium problems, our algorithm does not require any convexity or monotonicity conditions on the resulting bifunction. Some numerical experiments and comparisons are given to illustrate the efficiency of the proposed algorithm.An iterative method for equilibrium and constrained convex minimization problemshttps://zbmath.org/1491.470732022-09-13T20:28:31.338867Z"Yazdi, Maryam"https://zbmath.org/authors/?q=ai:yazdi.maryam"Shabani, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:shabani.mohammad-mehdi"Sababe, Saeed Hashemi"https://zbmath.org/authors/?q=ai:hashemi-sababe.saeedSummary: We are concerned with finding a common solution to an equilibrium problem associated with a bifunction, and a constrained convex minimization problem. We propose an iterative fixed point algorithm and prove that the algorithm generates a sequence strongly convergent to a common solution. The common solution is identified as the unique solution of a certain variational inequality.The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaceshttps://zbmath.org/1491.470742022-09-13T20:28:31.338867Z"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: The purpose of this paper is to propose and study an algorithm for solving the general split equality problem governed by Bregman quasi-nonexpansive mappings in Banach spaces. Under some mild conditions, we established the norm convergence of the proposed algorithm. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.Symplectic pseudospectral methods for optimal control. Theory and applications in path planninghttps://zbmath.org/1491.490012022-09-13T20:28:31.338867Z"Wang, Xinwei"https://zbmath.org/authors/?q=ai:wang.xinwei"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie|liu.jie.7|liu.jie.5|liu.jie.2|liu.jie.4|liu.jie.1|liu.jie.3"Peng, Haijun"https://zbmath.org/authors/?q=ai:peng.haijunThe book presents -- original, efficient, with good accuracy and convergence -- robust numerical methods for solving different nonlinear optimal control problems with dynamics described by ODEs.The methods combine the results from analytic mechanics with optimal control theory. Namely, optimal control problems are transformed into Hamiltonian systems by the Pontryagin's Maximum Principle or the Variational Principle. The phase flow in Hamiltonian systems is a symplectic transformation. So then, symplectic conservative numerical methods are used to solve optimal control problems efficiently. In the book, a series of symplectic pseudospectral methods (SPMs) for different optimal control problems are developed.
The book is divided into three parts. In Part I a short review of the numerical methods for nonlinear control problems such as indirect, direct, hybrid and artificial-based methods are presented. Next, the mathematical formulation of problems studied in the book and the necessary knowledge on the Hamiltonian structure of optimal control problems, symplectic theory and pseudospectral methods are discussed.
In Part II symplectic pseudospectral methods for nonlinear optimal control problems such as: unconstrained control problems, problems with inequality constraints, time-delayed problems, predictive control problems are developed.
In Part III interesting applications in trajectory planning and control related to: optimal maneuver of the spacecraft, trajectory planning for unmanned ground systems with different configurations, control of three-dimensional overhead cranes, trajectory planning for tractor-trailer systems are presented and discussed in details.
Examples in the book are shown according to the following scheme: problem formulation, analytic solution, if available, treated as a benchmark for SMP numerical solution, plots presenting both analytic and numerical solutions, discussion and remarks. Results of numerical simulations verify good properties of the presented methods.
In the book, the recent progress in symplectic numerical algorithms using pseudospectal methods for optimal control problems achieved by the authors is reported. The book is well-written. It can be recommended to researchers, engineers and graduate students who are interested in automatic control, optimal control problems and their effective solutions.
Reviewer: Wiesław Kotarski (Sosnowiec)Adjoint-based SQP method with block-wise quasi-Newton Jacobian updates for nonlinear optimal controlhttps://zbmath.org/1491.490152022-09-13T20:28:31.338867Z"Hespanhol, Pedro"https://zbmath.org/authors/?q=ai:hespanhol.pedro"Quirynen, Rien"https://zbmath.org/authors/?q=ai:quirynen.rienSummary: Nonlinear model predictive control (NMPC) generally requires the solution of a non-convex dynamic optimization problem at each sampling instant under strict timing constraints, based on a set of differential equations that can often be stiff and/or that may include implicit algebraic equations. This paper provides a local convergence analysis for the recently proposed adjoint-based sequential quadratic programming (SQP) algorithm that is based on a block-structured variant of the two-sided rank-one (TR1) quasi-Newton update formula to efficiently compute Jacobian matrix approximations in a sparsity preserving fashion. A particularly efficient algorithm implementation is proposed in case an implicit integration scheme is used for discretization of the optimal control problem, in which matrix factorization and matrix-matrix operations can be avoided entirely. The convergence analysis results as well as the computational performance of the proposed optimization algorithm are illustrated for two simulation case studies of NMPC.Optimal shape of stellarators for magnetic confinement fusionhttps://zbmath.org/1491.490312022-09-13T20:28:31.338867Z"Privat, Yannick"https://zbmath.org/authors/?q=ai:privat.yannick"Robin, Rémi"https://zbmath.org/authors/?q=ai:robin.remi"Sigalotti, Mario"https://zbmath.org/authors/?q=ai:sigalotti.marioSummary: We are interested in the design of stellarators, devices for the production of controlled nuclear fusion reactions alternative to tokamaks. The confinement of the plasma is entirely achieved by a helical magnetic field created by the complex arrangement of coils fed by high currents around a toroidal domain. Such coils describe a surface called ``coil winding surface'' (CWS). In this paper, we model the design of the CWS as a shape optimization problem, so that the cost functional reflects both optimal plasma confinement properties, through a least square discrepancy, and also manufacturability, thanks to geometrical terms involving the lateral surface or the curvature of the CWS.
We completely analyze the resulting problem: on the one hand, we establish the existence of an optimal shape, prove the shape differentiability of the criterion, and provide the expression of the differential in a workable form. On the other hand, we propose a numerical method and perform simulations of optimal stellarator shapes. We discuss the efficiency of our approach with respect to the literature in this area.Weak and approximate curvatures of a measure: a varifold perspectivehttps://zbmath.org/1491.490322022-09-13T20:28:31.338867Z"Buet, Blanche"https://zbmath.org/authors/?q=ai:buet.blanche"Leonardi, Gian Paolo"https://zbmath.org/authors/?q=ai:leonardi.gian-paolo"Masnou, Simon"https://zbmath.org/authors/?q=ai:masnou.simonSummary: By revisiting the notion of generalized second fundamental form originally introduced by Hutchinson for a special class of integral varifolds, we define a \textit{weak curvature tensor} that is particularly well-suited for being extended to general varifolds of any dimension and codimension through regularization. The resulting \textit{approximate second fundamental forms} are defined not only for piecewise-smooth surfaces, but also for datasets of very general type (like, e.g., point clouds). We obtain explicitly computable formulas for both weak and approximate curvature tensors, we exhibit structural properties and prove convergence results, and lastly we provide some numerical tests on point clouds that confirm the generality and effectiveness of our approach.A minimum-time obstacle-avoidance path planning algorithm for unmanned aerial vehicleshttps://zbmath.org/1491.490372022-09-13T20:28:31.338867Z"De Marinis, Arturo"https://zbmath.org/authors/?q=ai:de-marinis.arturo"Iavernaro, Felice"https://zbmath.org/authors/?q=ai:iavernaro.felice"Mazzia, Francesca"https://zbmath.org/authors/?q=ai:mazzia.francescaSummary: In this article, we present a new strategy to determine an unmanned aerial vehicle trajectory that minimizes its flight time in presence of avoidance areas and obstacles. The method combines classical results from optimal control theory, i.e. the Euler-Lagrange Theorem and the Pontryagin Minimum Principle, with a continuation technique that dynamically adapts the solution curve to the presence of obstacles. We initially consider the two-dimensional path planning problem and then move to the three-dimensional one, and include numerical illustrations for both cases to show the efficiency of our approach.Approximations of the connection Laplacian spectrahttps://zbmath.org/1491.580072022-09-13T20:28:31.338867Z"Burago, Dmitri"https://zbmath.org/authors/?q=ai:burago.dmitri"Ivanov, Sergei"https://zbmath.org/authors/?q=ai:ivanov.sergei-vladimirovich"Kurylev, Yaroslav"https://zbmath.org/authors/?q=ai:kurylev.yaroslav-v"Lu, Jinpeng"https://zbmath.org/authors/?q=ai:lu.jinpengBriefly, the authors focus their study on a convolution-type operator on vector bundles over metric-measure spaces. Also, they state that the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.
Reviewer: Mohammed El Aïdi (Bogotá)Sequential construction and dimension reduction of Gaussian processes under inequality constraintshttps://zbmath.org/1491.600512022-09-13T20:28:31.338867Z"Bachoc, François"https://zbmath.org/authors/?q=ai:bachoc.francois"López-Lopera, Andrés F."https://zbmath.org/authors/?q=ai:lopez-lopera.andres-f"Roustant, Olivier"https://zbmath.org/authors/?q=ai:roustant.olivierA unification of weighted and unweighted particle filtershttps://zbmath.org/1491.600542022-09-13T20:28:31.338867Z"Abedi, Ehsan"https://zbmath.org/authors/?q=ai:abedi.ehsan"Surace, Simone Carlo"https://zbmath.org/authors/?q=ai:surace.simone-carlo"Pfister, Jean-Pascal"https://zbmath.org/authors/?q=ai:pfister.jean-pascalExact simulation of two-parameter Poisson-Dirichlet random variableshttps://zbmath.org/1491.600702022-09-13T20:28:31.338867Z"Dassios, Angelos"https://zbmath.org/authors/?q=ai:dassios.angelos"Zhang, Junyi"https://zbmath.org/authors/?q=ai:zhang.junyiSummary: Consider a random vector \((V_1,\dots,V_n)\) where \(\{V_k\}_{k=1,\dots,n}\) are the first \(n\) components of a two-parameter Poisson-Dirichlet distribution \(PD(\alpha,\theta)\). In this paper, we derive a decomposition for the components of the random vector, and propose an exact simulation algorithm to sample from the random vector. Moreover, a special case arises when \(\theta/\alpha\) is a positive integer, for which we present a very fast modified simulation algorithm using a compound geometric representation of the decomposition. Numerical examples are provided to illustrate the accuracy and effectiveness of our algorithms.A high order weak approximation for jump-diffusions using Malliavin calculus and operator splittinghttps://zbmath.org/1491.600782022-09-13T20:28:31.338867Z"Akiyama, Naho"https://zbmath.org/authors/?q=ai:akiyama.naho"Yamada, Toshihiro"https://zbmath.org/authors/?q=ai:yamada.toshihiroSummary: The paper introduces a novel high order discretization scheme for expectation of jump-diffusion processes by using a Malliavin calculus approach and an operator splitting method. The test function of the target expectation is assumed to be only Lipschitz continuous in order to apply the method to financial problems. Then Kusuoka's estimate is employed to justify the proposed discretization scheme. The algorithm with a numerical example is shown for implementation.Strong convergence of split-step forward methods for stochastic differential equations driven by \(S \alpha S\) processeshttps://zbmath.org/1491.600892022-09-13T20:28:31.338867Z"Tarami, Bahram"https://zbmath.org/authors/?q=ai:tarami.bahram"Avaji, Mohsen"https://zbmath.org/authors/?q=ai:avaji.mohsenSummary: We consider stochastic differential equation driven by \(\alpha\)-stable processes. Three methods of drifting split-step Euler, diffused split-step Euler and three-stage Milstein for approximation of solutionare used. The strong convergence of these three methods is proven and the upper bounds of their stabilities are obtained and depicted.An efficient jet marcher for computing the quasipotential for 2D SDEs. Enhancing accuracy and efficiency of quasipotential solvershttps://zbmath.org/1491.601062022-09-13T20:28:31.338867Z"Paskal, Nicholas"https://zbmath.org/authors/?q=ai:paskal.nicholas"Cameron, Maria"https://zbmath.org/authors/?q=ai:cameron.maria-kourkinaSummary: We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths, EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet-Reygner sharp estimate for the expected escape time for the Maier-Stein SDE. Our codes are available on GitHub.Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemeshttps://zbmath.org/1491.601092022-09-13T20:28:31.338867Z"van Neerven, Jan"https://zbmath.org/authors/?q=ai:van-neerven.jan-m-a-m"Veraar, Mark"https://zbmath.org/authors/?q=ai:veraar.mark-cSummary: We prove a new Burkholder-Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if \((S(t,s))_{0\leq s\leq t\leq T}\) is a \(C_0\)-evolution family of contractions on a 2-smooth Banach space \(X\) and \((W_t)_{t\in [0,T]}\) is a cylindrical Brownian motion on a probability space \((\Omega, \mathbb{P})\) adapted to some given filtration, then for every \(0<p<\infty\) there exists a constant \(C_{p,X}\) such that for all progressively measurable processes \(g: [0,T]\times \Omega \rightarrow X\) the process \((\int_0^t S(t,s) g_s \,\text{d}W_s)_{t\in [0,T]}\) has a continuous modification and
\[
\mathbb{E}\sup_{t\in [0,T]}\Big\Vert \int_0^t S(t,s)g_s\,\text{d}W_s \Big\Vert^p\leq C_{p,X}^p \mathbb{E} \Big(\int_0^T \Vert g_t\Vert^2_{\gamma (H,X)}\,\text{d}t\Big)^{p/2}.
\]
Moreover, for \(2\leq p<\infty\) one may take \(C_{p,X} = 10 D \sqrt{p}\), where \(D\) is the constant in the definition of 2-smoothness for \(X\). The order \(O(\sqrt{p})\) coincides with that of Burkholder's inequality and is therefore optimal as \(p\rightarrow \infty\). Our result improves and unifies several existing maximal estimates and is even new in case \(X\) is a Hilbert space. Similar results are obtained if the driving martingale \(g_t\,\text{d} W_t\) is replaced by more general \(X\)-valued martingales \(\text{d} M_t\). Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs
\[
\text{d} u_t = A(t)u_t\,\text{d}t + g_t\,\text{d} W_t, \quad u_0 = 0,
\]
where the family \((A(t))_{t\in [0,T]}\) is assumed to generate a \(C_0\)-evolution family \((S(t,s))_{0\leq s\leq t\leq T}\) of contractions on a 2-smooth Banach spaces \(X\). Under spatial smoothness assumptions on the inhomogeneity \(g\), contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting diffusionshttps://zbmath.org/1491.601132022-09-13T20:28:31.338867Z"Jourdain, Benjamin"https://zbmath.org/authors/?q=ai:jourdain.benjamin"Tse, Alvin"https://zbmath.org/authors/?q=ai:tse.alvinSummary: In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivative. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. We use this result to deal with the contribution of the initialisation in the convergence of the fluctuations between the empirical measure of interacting diffusion and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. A complementary contribution related to the time evolution is treated using the \textit{master equation}, a parabolic PDE involving \(L\)-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.Explicit stabilized multirate method for stiff stochastic differential equationshttps://zbmath.org/1491.601152022-09-13T20:28:31.338867Z"Abdulle, Assyr"https://zbmath.org/authors/?q=ai:abdulle.assyr"Rosilho de Souza, Giacomo"https://zbmath.org/authors/?q=ai:de-souza.giacomo-rosilhoShort communication: A Gaussian Kusuoka approximation without solving random ODEshttps://zbmath.org/1491.601162022-09-13T20:28:31.338867Z"Yamada, Toshihiro"https://zbmath.org/authors/?q=ai:yamada.toshihiroConvergence rates of attractive-repulsive MCMC algorithmshttps://zbmath.org/1491.601222022-09-13T20:28:31.338867Z"Jiang, Yu Hang"https://zbmath.org/authors/?q=ai:jiang.yuhang"Liu, Tong"https://zbmath.org/authors/?q=ai:liu.tong.1|liu.tong"Lou, Zhiya"https://zbmath.org/authors/?q=ai:lou.zhiya"Rosenthal, Jeffrey S."https://zbmath.org/authors/?q=ai:rosenthal.jeffrey-s"Shangguan, Shanshan"https://zbmath.org/authors/?q=ai:shangguan.shanshan"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.2|wang.fei.1"Wu, Zixuan"https://zbmath.org/authors/?q=ai:wu.zixuanSummary: We consider MCMC algorithms for certain particle systems which include both attractive and repulsive forces, making their convergence analysis challenging. We prove that a version of these algorithms on a bounded state space is uniformly ergodic with explicit quantitative convergence rate. We also prove that a version on an unbounded state space is still geometrically ergodic, and then use the method of shift-coupling to obtain an explicit quantitative bound on its convergence rate.The policy iteration algorithm for a compound Poisson process applied to optimal dividend strategies under a Cramér-Lundberg risk modelhttps://zbmath.org/1491.601272022-09-13T20:28:31.338867Z"Liu, Guoxin"https://zbmath.org/authors/?q=ai:liu.guoxin"Liu, Xiaoying"https://zbmath.org/authors/?q=ai:liu.xiaoying"Liu, Zhaoyang"https://zbmath.org/authors/?q=ai:liu.zhaoyangSummary: In this paper, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem under the Cramér-Lundberg risk model. We conclude that the optimal value function is the minimum nonnegative solution of an optimization equation. Under any conditions, it can be approximated by iteration starting with the initial zero-valued policy, i.e., the policy of no dividend payment at all. An auxiliary optimization problem of maximizing dividend payment with terminal reward up to the first claim arrival time is introduced. This auxiliary optimization problem is solved completely in the sense that, no matter what the claim-size distribution is, the explicit formulae for both optimal strategy and value function are presented in general situation. These ensure that, at each iteration, we can get the explicit formulae of the improved strategy and its evaluation function. Finally, the PIA is illustrated with some numerical examples.A survey of sequential Monte Carlo methods for economics and financehttps://zbmath.org/1491.620082022-09-13T20:28:31.338867Z"Creal, Drew"https://zbmath.org/authors/?q=ai:creal.drew-dSummary: This article serves as an introduction and survey for economists to the field of sequential Monte Carlo methods which are also known as particle filters. Sequential Monte Carlo methods are simulation-based algorithms used to compute the high-dimensional and/or complex integrals that arise regularly in applied work. These methods are becoming increasingly popular in economics and finance; from dynamic stochastic general equilibrium models in macro-economics to option pricing. The objective of this article is to explain the basics of the methodology, provide references to the literature, and cover some of the theoretical results that justify the methods in practice.Marginal likelihood estimation with the cross-entropy methodhttps://zbmath.org/1491.621932022-09-13T20:28:31.338867Z"Chan, Joshua C. C."https://zbmath.org/authors/?q=ai:chan.joshua-c-c"Eisenstat, Eric"https://zbmath.org/authors/?q=ai:eisenstat.ericSummary: We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating \textit{independent} draws instead of \textit{correlated} MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is grounded in information theory, and therefore, is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications, the proposed CE method compares favorably to existing estimators.Bayesian analysis of instrumental variable models: acceptance-rejection within direct Monte Carlohttps://zbmath.org/1491.622712022-09-13T20:28:31.338867Z"Zellner, Arnold"https://zbmath.org/authors/?q=ai:zellner.arnold"Ando, Tomohiro"https://zbmath.org/authors/?q=ai:ando.tomohiro"Baştürk, Nalan"https://zbmath.org/authors/?q=ai:basturk.nalan"Hoogerheide, Lennart"https://zbmath.org/authors/?q=ai:hoogerheide.lennart-f"van Dijk, Herman K."https://zbmath.org/authors/?q=ai:van-dijk.herman-kSummary: We discuss Bayesian inferential procedures within the family of instrumental variables regression models and focus on two issues: existence conditions for posterior moments of the parameters of interest under a flat prior and the potential of Direct Monte Carlo (DMC) approaches for efficient evaluation of such possibly highly non-elliptical posteriors. We show that, for the general case of \(m\) endogenous variables under a flat prior, posterior moments of order \(r\) exist for the coefficients reflecting the endogenous regressors' effect on the dependent variable, if the number of instruments is greater than \(m +r\), even though there is an issue of local non-identification that causes non-elliptical shapes of the posterior. This stresses the need for efficient Monte Carlo integration methods. We introduce an extension of DMC that incorporates an acceptance-rejection sampling step within DMC. This Acceptance-Rejection within Direct Monte Carlo (ARDMC) method has the attractive property that the generated random drawings are independent, which greatly helps the fast convergence of simulation results, and which facilitates the evaluation of the numerical accuracy. The speed of ARDMC can be easily further improved by making use of parallelized computation using multiple core machines or computer clusters. We note that ARDMC is an analogue to the well-known ``Metropolis-Hastings within Gibbs'' sampling in the sense that one `more difficult' step is used within an `easier' simulation method. We compare the ARDMC approach with the Gibbs sampler using simulated data and two empirical data sets, involving the settler mortality instrument of \textit{D. Acemoglu} et al. [``The colonial origins of comparative development: an empirical investigation'', Am. Econ. Rev. 91, No. 5, 1369--1401 (2001; \url{doi:10.1257/aer.91.5.1369})] and father's education's instrument used by \textit{L. Hoogerheide} et al. [``Family background variables as instruments for education in income regressions: a Bayesian analysis'', Econ. Educ. Rev. 31, No. 5, 515--523 (2012; \url{doi:10.1016/j.econedurev.2012.03.001})]. Even without making use of parallelized computation, an efficiency gain is observed both under strong and weak instruments, where the gain can be enormous in the latter case.Handbook of fractional calculus for engineering and sciencehttps://zbmath.org/1491.650012022-09-13T20:28:31.338867ZPublisher's description: Fractional calculus is used to model many real-life situations from science and engineering. The book includes different topics associated with such equations and their relevance and significance in various scientific areas of study and research. In this book readers will find several important and useful methods and techniques for solving various types of fractional-order models in science and engineering. The book should be useful for graduate students, PhD students, researchers and educators interested in mathematical modelling, physical sciences, engineering sciences, applied mathematical sciences, applied sciences, and so on.
This Handbook:
\begin{itemize}
\item Provides reliable methods for solving fractional-order models in science and engineering.
\item Contains efficient numerical methods and algorithms for engineering-related equations.
\item Contains comparison of various methods for accuracy and validity.
\item Demonstrates the applicability of fractional calculus in science and engineering.
\item Examines qualitative as well as quantitative properties of solutions of various types of science- and engineering-related equations.
\end{itemize}
Readers will find this book to be useful and valuable in increasing and updating their knowledge in this field and will be it will be helpful for engineers, mathematicians, scientist and researchers working on various real-life problems.
The articles of this volume will not be indexed individually.Exploring Monte Carlo methodshttps://zbmath.org/1491.650022022-09-13T20:28:31.338867Z"Dunn, William L."https://zbmath.org/authors/?q=ai:dunn.william-l"Shultis, J. Kenneth"https://zbmath.org/authors/?q=ai:shultis.j-kennethPublisher's description: Exploring Monte Carlo Methods, Second Edition provides a valuable introduction to the numerical methods that have come to be known as ``Monte Carlo''. This unique and trusted resource for course use, as well as researcher reference, offers accessible coverage, clear explanations and helpful examples throughout. Building from the basics, the text also includes applications in a variety of fields, such as physics, nuclear engineering, finance and investment, medical modeling and prediction, archaeology, geology and transportation planning.
For the first edition see [Zbl 1261.65006].Numerical analysis and scientific computationhttps://zbmath.org/1491.650032022-09-13T20:28:31.338867Z"Leader, Jeffery J."https://zbmath.org/authors/?q=ai:leader.jeffery-jPublisher's description: This is an introductory single-term numerical analysis text with a modern scientific computing flavor. It offers an immediate immersion in numerical methods featuring an up-to-date approach to computational matrix algebra and an emphasis on methods used in actual software packages, always highlighting how hardware concerns can impact the choice of algorithm. It fills the need for a text that is mathematical enough for a numerical analysis course yet applied enough for students of science and engineering taking it with practical need in mind.
The standard methods of numerical analysis are rigorously derived with results stated carefully and many proven. But while this is the focus, topics such as parallel implementations, the Basic Linear Algebra Subroutines, halfto quadruple-precision computing, and other practical matters are frequently discussed as well.
Prior computing experience is not assumed. Optional MATLAB subsections for each section provide a comprehensive self-taught tutorial and also allow students to engage in numerical experiments with the methods they have just read about. The text may also be used with other computing environments.
This new edition offers a complete and thorough update. Parallel approaches, emerging hardware capabilities, computational modeling, and data science are given greater weight.Finite element method. Physics and solution methodshttps://zbmath.org/1491.650042022-09-13T20:28:31.338867Z"Müftü, Sinan"https://zbmath.org/authors/?q=ai:muftu.sinanPublisher's description: Finite Element Method: Physics and Solution Methods aims to provide the reader a sound understanding of the physical systems and solution methods to enable effective use of the finite element method. This book focuses on one- and two-dimensional elasticity and heat transfer problems with detailed derivations of the governing equations. The connections between the classical variational techniques and the finite element method are carefully explained. Following the chapter addressing the classical variational methods, the finite element method is developed as a natural outcome of these methods where the governing partial differential equation is defined over a subsegment (element) of the solution domain. As well as being a guide to thorough and effective use of the finite element method, this book also functions as a reference on theory of elasticity, heat transfer, and mechanics of beams.Sparse grids and applications -- Munich 2018. Selected papers based on the presentations at the fifth workshop, SGA2018, Munich, Germany, July 23--27, 2018https://zbmath.org/1491.650052022-09-13T20:28:31.338867ZPublisher's description: Sparse grids are a popular tool for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different flavors, are frequently the method of choice.
This volume of LNCSE presents selected papers from the proceedings of the fifth workshop on sparse grids and applications, and demonstrates once again the importance of this numerical discretization scheme. The articles present recent advances in the numerical analysis of sparse grids in connection with a range of applications including uncertainty quantification, plasma physics simulations, and computational chemistry, to name but a few.
The articles of this volume will be reviewed individually. For the preceding workshop see [Zbl 1397.65009].Proceedings of the seventh international conference on mathematics and computing, ICMC 2021, Shibpur, India, March 2--5, 2021https://zbmath.org/1491.650062022-09-13T20:28:31.338867ZPublisher's description: This book features selected papers from the 7th International Conference on Mathematics and Computing (ICMC 2021), organized by Indian Institute of Engineering Science and Technology (IIEST), Shibpur, India, during March 2021. It covers recent advances in the field of mathematics, statistics, and scientific computing. The book presents innovative work by leading academics, researchers, and experts from industry.
The articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1468.65003].Monte Carlo with determinantal point processeshttps://zbmath.org/1491.650072022-09-13T20:28:31.338867Z"Bardenet, Rémi"https://zbmath.org/authors/?q=ai:bardenet.remi"Hardy, Adrien"https://zbmath.org/authors/?q=ai:hardy.adrienSummary: We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical \(N^{-1/2}\), where \(N\) is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as \(N^{-(1+1/d)/2}\), where \(d\) is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition; a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.Efficient algorithms for tail probabilities of exchangeable lognormal sumshttps://zbmath.org/1491.650082022-09-13T20:28:31.338867Z"Dingeç, Kemal Dinçer"https://zbmath.org/authors/?q=ai:dingec.kemal-dincer"Hörmann, Wolfgang"https://zbmath.org/authors/?q=ai:hormann.wolfgangSummary: For the estimation of left and right-tail probabilities and the pdf of the sum of exchangeable lognormal random vectors a new conditional Monte Carlo (CMC) algorithm is developed. It removes the randomness of the sum of all input variables and is simple and fast. For estimating the left-tail probabilities the CMC algorithm is logarithmically efficient. A further improvement of the algorithm by removing also the randomness of the radius of the normal input using close to optimal one dimensional importance sampling, results in the CMC.RCMC algorithm. For the sum of independent and identically distributed (i.i.d.) and exchangeable lognormal vectors it is the first algorithm that has bounded relative error for the left-tail probabilities. The CMC.RCMC algorithm is logarithmically efficient for the right-tail probabilities. Numerical experiments verify that it has a very good performance for all left-tail estimation problems and a good performance for the right tail for probabilities not smaller than \(10^{-10}\). When estimating the pdf the relative errors observed are all very close to those of the corresponding probability estimates.Analysis of Langevin Monte Carlo via convex optimizationhttps://zbmath.org/1491.650092022-09-13T20:28:31.338867Z"Durmus, Alain"https://zbmath.org/authors/?q=ai:durmus.alain"Majewski, Szymon"https://zbmath.org/authors/?q=ai:majewski.szymon"Miasojedow, Błażej"https://zbmath.org/authors/?q=ai:miasojedow.blazejSummary: In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as the first order optimization algorithm for an objective functional defined on the Wasserstein space of order 2. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from log-concave smooth target distribution on \(\mathbb{R}^d\). Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution. These new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm for largescale Bayesian inference. Using the optimization perspective, we provide non-asymptotic convergence analysis for the newly proposed methods.Analysis of a multilevel Markov chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusionshttps://zbmath.org/1491.650102022-09-13T20:28:31.338867Z"Hoang, Viet Ha"https://zbmath.org/authors/?q=ai:hoang.viet-ha"Quek, Jia Hao"https://zbmath.org/authors/?q=ai:quek.jia-hao"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophUnbiased simulation of rare events in continuous timehttps://zbmath.org/1491.650112022-09-13T20:28:31.338867Z"Hodgson, James"https://zbmath.org/authors/?q=ai:hodgson.james"Johansen, Adam M."https://zbmath.org/authors/?q=ai:johansen.adam-m"Pollock, Murray"https://zbmath.org/authors/?q=ai:pollock.murraySummary: For rare events described in terms of Markov processes, truly unbiased estimation of the rare event probability generally requires the avoidance of numerical approximations of the Markov process. Recent work in the exact and \(\varepsilon \)-strong simulation of diffusions, which can be used to almost surely constrain sample paths to a given tolerance, suggests one way to do this. We specify how such algorithms can be combined with the classical multilevel splitting method for rare event simulation. This provides unbiased estimations of the probability in question. We discuss the practical feasibility of the algorithm with reference to existing \(\varepsilon \)-strong methods and provide proof-of-concept numerical examples.About one method of numerical solution Schrödinger equationhttps://zbmath.org/1491.650122022-09-13T20:28:31.338867Z"Plokhotnikov, K. E."https://zbmath.org/authors/?q=ai:plokhotnikov.k-ehSummary: The paper considers the method of numerical solution of the Schrödinger equation, which, in part, can be attributed to the class of Monte Carlo methods. The method is presented and simultaneously illustrated by the examples of solving the one-dimensional and multidimensional Schrödinger equation in the problems of linear one-dimensional oscillator, hydrogen atom and benzene.A method of generating random vectors with a given probability density functionhttps://zbmath.org/1491.650132022-09-13T20:28:31.338867Z"Darkhovsky, B. S."https://zbmath.org/authors/?q=ai:darkhovsky.boris-s"Popkov, Yu. S."https://zbmath.org/authors/?q=ai:popkov.yuri-s"Popkov, A. Yu."https://zbmath.org/authors/?q=ai:popkov.alexey-yu"Aliev, A. S."https://zbmath.org/authors/?q=ai:alyev.a-sSummary: We propose a method for generating random independent vectors that have a given continuous distribution density with compact support. The main advantage of the proposed method are guaranteed estimates of the error in the generation of random vectors. We show an illustrative experimental comparison of the proposed method with the Metropolis-Hastings algorithm.A splitting method for SDEs with locally Lipschitz drift: illustration on the FitzHugh-Nagumo modelhttps://zbmath.org/1491.650142022-09-13T20:28:31.338867Z"Buckwar, Evelyn"https://zbmath.org/authors/?q=ai:buckwar.evelyn"Samson, Adeline"https://zbmath.org/authors/?q=ai:samson.adeline"Tamborrino, Massimiliano"https://zbmath.org/authors/?q=ai:tamborrino.massimiliano"Tubikanec, Irene"https://zbmath.org/authors/?q=ai:tubikanec.ireneSummary: In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.Solving elliptic equations with Brownian motion: bias reduction and temporal difference learninghttps://zbmath.org/1491.650152022-09-13T20:28:31.338867Z"Martin, Cameron"https://zbmath.org/authors/?q=ai:martin.cameron"Zhang, Hongyuan"https://zbmath.org/authors/?q=ai:zhang.hongyuan"Costacurta, Julia"https://zbmath.org/authors/?q=ai:costacurta.julia"Nica, Mihai"https://zbmath.org/authors/?q=ai:nica.mihai.2|nica.mihai"Stinchcombe, Adam R."https://zbmath.org/authors/?q=ai:stinchcombe.adam-rSummary: The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from issues related to statistical bias and sampling efficiency. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.An adaptive time-stepping method based on a posteriori weak error analysis for large SDE systemshttps://zbmath.org/1491.650162022-09-13T20:28:31.338867Z"Merle, Fabian"https://zbmath.org/authors/?q=ai:merle.fabian"Prohl, Andreas"https://zbmath.org/authors/?q=ai:prohl.andreasIn this paper, a numerical scheme based on the semi-implicit Euler method is constructed to approximate solutions of a class of systems of stochastic differential equations. An adaptive mesh strategy (using quasi deterministic rather than random meshes) is developed in which local step size is chosen using an a posteriori weak error estimator.
The authors prove convergence with optimal weak order for the estimator and an algorithm for weak adaptive approximation is presented. Then they prove the number of iterations necessary to generate each local step size and the number of time steps needed to cover the desired finite interval. The paper concludes with results of numerical experiments that illustrate advantages of this adaptive strategy over a uniform step size strategy.
Reviewer: Melvin D. Lax (Long Beach)Asymptotic analysis in multivariate average case approximation with Gaussian kernelshttps://zbmath.org/1491.650172022-09-13T20:28:31.338867Z"Khartov, A. A."https://zbmath.org/authors/?q=ai:khartov.aleksei-andreevich"Limar, I. A."https://zbmath.org/authors/?q=ai:limar.i-aSummary: We consider tensor product random fields \(Y_d\), \(d \in \mathbb{N} \), whose covariance functions are Gaussian kernels with a given sequence of length scale parameters. We investigate the growth of the average case approximation complexity \(n^{Y_d}(\varepsilon)\) of \(Y_d\) for arbitrary fixed \(\varepsilon \in(0, 1)\) and \(d \to \infty \). Namely, we find criteria for the boundedness of \(n^{Y_d}(\varepsilon)\) depending on \(d\) and criteria for \(n^{Y_d}(\varepsilon) \to \infty\) when \(d \to \infty\) for any fixed \(\varepsilon \in(0, 1)\). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics
\[
\ln n^{Y_d} (\varepsilon) = a_d + q (\varepsilon) b_d + o (b_d), \quad d \to \infty,
\]
with any \(\varepsilon \in(0, 1)\). Here \(q :(0, 1) \to \mathbb{R}\) is a non-decreasing function, \((a_d)_{d \in \mathbb{N}}\) is a sequence and \((b_d)_{d \in \mathbb{N}}\) is a positive sequence such that \(b_d \to \infty\), \(d \to \infty \). We show that only special quantiles of self-decomposable distribution functions appear as functions \(q\) in a given asymptotics. These general results apply to \(n^{Y_d}(\varepsilon)\) under particular assumptions on the length scale parameters.Solving the interference problem for ellipses and ellipsoids: new formulaehttps://zbmath.org/1491.650182022-09-13T20:28:31.338867Z"Caravantes, J."https://zbmath.org/authors/?q=ai:caravantes.jorge"Diaz-Toca, G. M."https://zbmath.org/authors/?q=ai:diaz-toca.gema-maria"Fioravanti, M."https://zbmath.org/authors/?q=ai:fioravanti.mario-a"Gonzalez-Vega, L."https://zbmath.org/authors/?q=ai:gonzalez-vega.laureanoEllipses and ellipsoids are essential entities for modelling (and/or enclosing) the shape of the objects under consideration. So, in this sense, the problem of detecting when two moving ellipses or ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc.
In this paper, the authors consider the characteristic polynomial of the pencil defined by two ellipses/ellipsoids \(A\) and \(B\) given by \(X^T AX = 0\) and \(X^T BX = 0\) and they analyze symbolically the sign of the real roots of the characteristic polynomial. From this, new formulae are derived when \(A\) and \(B\) overlap, are separate, or touch each other externally. This characterization is defined by a minimal set of polynomial inequalities depending only on the entries of \(A\) and \(B\). Thus, one of the fascinating result of this paper is that one needs only to compute the characteristic polynomial of the pencil defined by \(A\) and \(B\), det\((TA + B)\), and not the intersection points between them.
The authors compare the results with the best available approach dealing with this problem, and they show that the new formulae involve a smaller set of polynomials and less sign conditions.
Finally, as an application, it is shown that, since the analysis of the univariate polynomials (depending on the time) in the formulae provides the collision events between them, the obtained characterization provides a new approach for exact collision detection of two moving ellipses or ellipsoids.
Reviewer: Sonia Pérez Díaz (Madrid)Curve approximation by adaptive neighborhood simulated annealing and piecewise Bézier curveshttps://zbmath.org/1491.650192022-09-13T20:28:31.338867Z"Ueda, E. K."https://zbmath.org/authors/?q=ai:ueda.e-k"Sato, A. K."https://zbmath.org/authors/?q=ai:sato.andre-kubagawa"Martins, T. C."https://zbmath.org/authors/?q=ai:martins.thiago-c"Takimoto, R. Y."https://zbmath.org/authors/?q=ai:takimoto.rogerio-y"Rosso, R. S. U. jun."https://zbmath.org/authors/?q=ai:rosso.r-s-u-jun"Tsuzuki, M. S. G."https://zbmath.org/authors/?q=ai:tsuzuki.marcos-s-gSummary: The curve approximation problem is widely researched in CAD/CAM and geometric modelling. The problem consists in determining an approximating curve from a given sequence of points. The usual approach is the minimization of the discrepancy between the approximating curve and the given sequence of points. However, the minimization of just the discrepancy leads to the overfitting problem, in which the solution is not unique. A new approach is proposed to overcome this problem, in which the length of the approximating curve is used as a regularization increasing the algorithm stability. Another new proposal is the discrepancy determination, in which a method that has the best ratio between accuracy and processing time is proposed. A new simulated annealing (SA) approach is used to minimize the problem, in which the next candidate is determined by a probability distribution controlled by the crystallization factor. The crystallization factor is low for higher temperatures ensuring the exploration of the domain. The crystallization factor is high for lower temperatures, corresponding the refinement phase of the SA. The approximating curve is represented as a piecewise cubic Bézier curve, which is a sequence of several connected cubic Bézier curves. The piecewise Bézier curve supports a new proposed data structure that improves the proposed algorithm. A comparison is also made between the used single-objective SA and the AMOSA multi-objective SA. The results showed that the proposed single-objective SA finds a solution which is not dominated by the Pareto front determined by AMOSA. The results also showed that the regularization stabilized the algorithm, in which the increase in parameters does not lead to the overfitting problem. The proposed algorithm can process even complex curves with self-intersections and higher curvature.Fractional derivative based nonlinear diffusion model for image denoisinghttps://zbmath.org/1491.650202022-09-13T20:28:31.338867Z"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh.3|kumar.santosh.4|kumar.santosh.2|kumar.santosh.1|kumar.santosh"Alam, Khursheed"https://zbmath.org/authors/?q=ai:alam.khursheed"Chauhan, Alka"https://zbmath.org/authors/?q=ai:chauhan.alkaSummary: In this article, a new conformable fractional anisotropic diffusion model for image denoising is presented, which contains the spatial derivative along with the time-fractional derivative. This model is a generalization of the diffusion model [\textit{M. Welk} et al., Lect. Notes Comput. Sci. 3459, 585--597 (2005; Zbl 1119.68511)] with forward-backward diffusivities. The proposed model is very efficient for noise removal of the noisy images in comparison to the classical anisotropic diffusion model. The numerical experiments are performed using an explicit scheme for different-different values of fractional order derivative \(\alpha \). The experimental results are obtained in terms of peak signal to noise ratio (PSNR) as a metric.The gamma function via interpolationhttps://zbmath.org/1491.650212022-09-13T20:28:31.338867Z"Causley, Matthew F."https://zbmath.org/authors/?q=ai:causley.matthew-fSummary: A new computational framework for evaluation of the gamma function \(\Gamma (z)\) over the complex plane is developed. The algorithm is based on interpolation by rational functions, and generalizes the classical methods of \textit{C. Lanczos} [J. Soc. Ind. Appl. Math., Ser. B, Numer. Anal. 1, 86--96 (1964; Zbl 0136.05201)] and \textit{J. L. Spouge} [SIAM J. Numer. Anal. 31, No. 3, 931--944 (1994; Zbl 0803.40001)] (which we show are also interpolatory). This framework utilizes the exact poles of the gamma function. By relaxing this condition and allowing the poles to vary, a near-optimal rational approximation is possible, which is demonstrated using the adaptive Antoulous Anderson (AAA) algorithm, developed in [\textit{Y. Nakatsukasa} et al., SIAM J. Sci. Comput. 40, No. 3, A1494--A1522 (2018; Zbl 1390.41015)] and [\textit{Y. Nakatsukasa} and \textit{L. N. Trefethen}, SIAM J. Sci. Comput. 42, No. 5, A3157--A3179 (2020; Zbl 1452.65035)]. The resulting approximations are competitive with Stirling's formula in terms of overall efficiency.Comparison of gradient approximation methods in schemes designed for scale-resolving simulationshttps://zbmath.org/1491.650222022-09-13T20:28:31.338867Z"Bakhné, S."https://zbmath.org/authors/?q=ai:bakhne.s"Bosnyakov, S. M."https://zbmath.org/authors/?q=ai:bosnyakov.s-m"Mikhaĭlov, S. V."https://zbmath.org/authors/?q=ai:mikhailov.s-v"Troshin, A. I."https://zbmath.org/authors/?q=ai:troshin.a-iSummary: Various methods for improved accuracy approximation of the gradients entering the diffusion fluxes are considered. Linear combinations of 2nd order difference schemes for a non-uniform grid that transform into 4th order schemes in the uniform case were investigated. We also considered 3rd and 4th order schemes for approximating gradients on a non-uniform grid in the normal and tangent directions to the cell face, respectively, based on Lagrange polynomials. The initial testing was carried out on one-dimensional functions: a smooth Gauss function and a piecewise linear function. Next, the schemes were applied in direct numerical simulation of the Taylor-Green vortex.On optimal adaptive quadratures for automatic integrationhttps://zbmath.org/1491.650232022-09-13T20:28:31.338867Z"Goćwin, Maciej"https://zbmath.org/authors/?q=ai:gocwin.maciejSummary: In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an \(r\) times differentiable function with the assumption that the \(r\) th derivative is positive within precision \(\varepsilon >0\) are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses the minimal number of function evaluations. It is also shown how to adapt the quadratures to the case where the assumption that the \(r\)th derivative has a constant sign is not fulfilled. The theoretical results are illustrated and confirmed with numerical tests.A numerical algorithm to inversing a Toeplitz heptadiagonal matrixhttps://zbmath.org/1491.650242022-09-13T20:28:31.338867Z"Talibi, B."https://zbmath.org/authors/?q=ai:talibi.b"Hadj, A. Aiat"https://zbmath.org/authors/?q=ai:hadj.a-aiat"Sarsri, D."https://zbmath.org/authors/?q=ai:sarsri.drissSummary: In this paper, we compare two algorithms for computing the inverse of a heptadiagonal Toeplitz matrix. The numerical results are given to compare the effectiveness of this method.Fast and improved scaled HSS preconditioner for steady-state space-fractional diffusion equationshttps://zbmath.org/1491.650252022-09-13T20:28:31.338867Z"Chen, Fang"https://zbmath.org/authors/?q=ai:chen.fang"Li, Tian-Yi"https://zbmath.org/authors/?q=ai:li.tianyiSummary: For the discrete linear system resulted from the considered steady-state space-fractional diffusion equations, we propose an improved scaled HSS (ISHSS) iteration method and discuss its convergence theory. Then, we construct a fast ISHSS (FISHSS) preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods. We discuss the spectral properties of the FISHSS preconditioning matrix. Numerical experiments show the good performance of the FISHSS preconditioner.Two-level Nyström-Schur preconditioner for sparse symmetric positive definite matriceshttps://zbmath.org/1491.650262022-09-13T20:28:31.338867Z"Daas, Hussam Al"https://zbmath.org/authors/?q=ai:daas.hussam-al"Rees, Tyrone"https://zbmath.org/authors/?q=ai:rees.tyrone"Scott, Jennifer"https://zbmath.org/authors/?q=ai:scott.jennifer-aOn the generalized AOR and CG iteration methods for a class of block two-by-two linear systemshttps://zbmath.org/1491.650272022-09-13T20:28:31.338867Z"Balani, Fariba Bakrani"https://zbmath.org/authors/?q=ai:balani.fariba-bakrani"Hajarian, Masoud"https://zbmath.org/authors/?q=ai:hajarian.masoudSummary: In this work, we utilize the generalized AOR (GAOR) and CG (GCG) methods for constructing iteration methods to solve the block two-by-two linear systems which arise from the solution of the complex symmetric linear systems of equations. In order to compare the GAOR and GCG methods with some existing methods, we present some numerical examples to illustrate the performance of these methods.A note on augmented unprojected Krylov subspace methodshttps://zbmath.org/1491.650282022-09-13T20:28:31.338867Z"Soodhalter, Kirk M."https://zbmath.org/authors/?q=ai:soodhalter.kirk-mSummary: Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [the author et al., ``A survey of subspace recycling iterative methods'', Mitt. Ges. Angew. Math. Mech. 43, No. 4, Article ID e202000016, 29 p. (2020; \url{doi:10.1002/gamm.202000016})]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the \(R^3\) GMRES method proposed in [\textit{Y. Dong} et al., ETNA, Electron. Trans. Numer. Anal. 42, 136--146 (2014; Zbl 1307.65049)] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [\textit{Y. Saad}, SIAM J. Matrix Anal. Appl. 18, No. 2, 435--449 (1997; Zbl 0871.65026)].On a new weight tri-diagonal iterative method and its applicationshttps://zbmath.org/1491.650292022-09-13T20:28:31.338867Z"Yambangwai, D."https://zbmath.org/authors/?q=ai:yambangwai.damrongsak"Cholamjiak, W."https://zbmath.org/authors/?q=ai:cholamjiak.watcharaporn"Thianwan, T."https://zbmath.org/authors/?q=ai:thianwan.tanakit"Dutta, H."https://zbmath.org/authors/?q=ai:dutta.hemenSummary: In this article, a new weight tri-diagonal iterative method in solving system of linear equation is proposed and its convergence is discussed. The set of methods in solving linear system generated by high-order scheme for solving one-dimensional Poisson equation and one-dimensional heat equation with periodic boundary are presented. The numerical experiments shows that the proposed method demonstrates a better performance compared with weight Jacobi, successive-over relaxation and alternating group explicit methods.A contribution to the conditioning of the mixed least-squares scaled total least-squares problemhttps://zbmath.org/1491.650302022-09-13T20:28:31.338867Z"Meng, Lingsheng"https://zbmath.org/authors/?q=ai:meng.lingshengSummary: A new closed formula for first-order perturbation estimates for the solution of the mixed least-squares scaled total least-squares (MLSSTLS) problem is presented. It is mathematically equivalent to the one by \textit{P. Zhang} and \textit{Q. Wang} [Numer. Algorithms 89, No. 3, 1223--1246 (2022; Zbl 1490.65065)]. With this formula, new closed formulas for the relative normwise, mixed, and componentwise condition numbers of the MLSSTLS problem are derived. Compact forms and upper bounds for the relative normwise condition number are also given in order to obtain economic storage and efficient computations.Random Toeplitz matrices: the condition number under high stochastic dependencehttps://zbmath.org/1491.650312022-09-13T20:28:31.338867Z"Manrique-Mirón, Paulo"https://zbmath.org/authors/?q=ai:manrique-miron.paulo-cesarResidual and restarting in Krylov subspace evaluation of the \(\varphi\) functionhttps://zbmath.org/1491.650322022-09-13T20:28:31.338867Z"Botchev, Mike A."https://zbmath.org/authors/?q=ai:botchev.mike-a"Knizhnerman, Leonid"https://zbmath.org/authors/?q=ai:knizhnerman.leonid"Tyrtyshnikov, Eugene E."https://zbmath.org/authors/?q=ai:tyrtyshnikov.evgenii-eDecay bounds for Bernstein functions of Hermitian matrices with applications to the fractional graph Laplacianhttps://zbmath.org/1491.650332022-09-13T20:28:31.338867Z"Schweitzer, Marcel"https://zbmath.org/authors/?q=ai:schweitzer.marcelSummary: For many functions of matrices \(f(A)\), it is known that their entries exhibit a rapid -- often exponential or even superexponential -- decay away from the sparsity pattern of the matrix \(A\). In this paper, we specifically focus on the class of Bernstein functions, which contains the fractional powers \(A^\alpha\), \(\alpha \in (0,1)\), as an important special case, and derive new decay bounds by exploiting known results for the matrix exponential in conjunction with the Lévy-Khintchine integral representation. As a particular special case, we find a result concerning the power law decay of the strength of connection in nonlocal network dynamics described by the fractional graph Laplacian, which improves upon known results from the literature by doubling the exponent in the power law.An ultrafast cellular method for matrix multiplicationhttps://zbmath.org/1491.650342022-09-13T20:28:31.338867Z"Jelfimova, L. D."https://zbmath.org/authors/?q=ai:jelfimova.l-dSummary: The author considers the ultrafast cellular method of matrix multiplication that deals with cellular submatrices, interacts with well-known matrix multiplication cellular methods, and minimizes by 12.5\% the computing complexity of cellular analogs of well-known matrix multiplication algorithms generated on their basis. The interaction of the ultrafast cellular method with the unified cellular method of matrix multiplication provides the highest (as compared with well-known methods) percentage (equal to 45.2\%) of minimization of multiplicative, additive, and overall complexities of well-known matrix multiplication algorithms. The computing complexity of the ultrafast method is estimated using the models of getting cellular analogs of the traditional matrix multiplication algorithm.An ATLD-ALS method for the trilinear decomposition of large third-order tensorshttps://zbmath.org/1491.650352022-09-13T20:28:31.338867Z"Simonacci, Violetta"https://zbmath.org/authors/?q=ai:simonacci.violetta"Gallo, Michele"https://zbmath.org/authors/?q=ai:gallo.micheleSummary: CP decomposition of large third-order tensors can be computationally challenging. Parameters are typically estimated by means of the ALS procedure because it yields least-squares solutions and provides consistent outcomes. Nevertheless, ALS presents two major flaws which are particularly problematic for large-scale problems: slow convergence and sensitiveness to degeneracy conditions such as over-factoring, collinearity, bad initialization and local minima. More efficient algorithms have been proposed in the literature. They are, however, much less dependable than ALS in delivering stable results because the increased speed often comes at the expense of accuracy. In particular, the ATLD procedure is one of the fastest alternatives, but it is hardly employed because of the unreliable nature of its convergence. As a solution, multi-optimization is proposed. ATLD and ALS steps are concatenated in an integrated procedure with the purpose of increasing efficiency without a significant loss in precision. This methodology has been implemented and tested under realistic conditions on simulated data sets.Extended Kung-Traub-type method for solving equationshttps://zbmath.org/1491.650362022-09-13T20:28:31.338867Z"Argyros, Ioannis K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"Santhosh, George"https://zbmath.org/authors/?q=ai:santhosh.georgeSummary: We are motivated by a Kung-Traub-type method for solving equations on the real line. In particular, we extend this method for Banach space valued operators. The radius of convergence is also obtained as well as error bounds on the distances involved and a uniqueness result. Our convergence analysis avoids Taylor expansions and the computation of higher order than one derivatives.Inertial parallel relaxed CQ algorithms for common split feasibility problems and its application in signal recoveryhttps://zbmath.org/1491.650372022-09-13T20:28:31.338867Z"Cholamjiak, Watcharaporn"https://zbmath.org/authors/?q=ai:cholamjiak.watcharaporn"Witthayarat, Uamporn"https://zbmath.org/authors/?q=ai:witthayarat.uamporn"Suparatulatorn, Raweerote"https://zbmath.org/authors/?q=ai:suparatulatorn.raweeroteSummary: We introduce new inertial algorithms together with the modified step size using linesearch technique for solving common split feasibility problems in real Hilbert spaces. We also establish a convergence analysis and achieve a weak convergence to the solution of the problems under some mild conditions. Moreover, we apply our results to solve the signal recovery problem and present two experiments in both of its effectiveness and the performance comparison to previous algorithms in the literature. Our result generalizes and extends to previous results involving the common split feasibility problem.A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaceshttps://zbmath.org/1491.650382022-09-13T20:28:31.338867Z"Jolaoso, Lateef Olakunle"https://zbmath.org/authors/?q=ai:jolaoso.lateef-olakunleSummary: In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.Deep null space learning for inverse problems: convergence analysis and rateshttps://zbmath.org/1491.650392022-09-13T20:28:31.338867Z"Schwab, Johannes"https://zbmath.org/authors/?q=ai:schwab.johannes"Antholzer, Stephan"https://zbmath.org/authors/?q=ai:antholzer.stephan"Haltmeier, Markus"https://zbmath.org/authors/?q=ai:haltmeier.markusStrong convergence of inertial subgradient extragradient algorithm for solving pseudomonotone equilibrium problemshttps://zbmath.org/1491.650402022-09-13T20:28:31.338867Z"Thong, Duong Viet"https://zbmath.org/authors/?q=ai:duong-viet-thong."Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasit"Rassias, Michael T."https://zbmath.org/authors/?q=ai:rassias.michael-th"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, we propose a new modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. We establish the strong convergence of the proposed method under several suitable conditions. In addition, the linear convergence is obained under strong pseudomonotonicity assumption. Our results generalize and extend some related results in the literature. Finally, we provide numerical experiments to illustrate the performance of the proposed algorithm.Two fast converging inertial subgradient extragradient algorithms with variable stepsizes for solving pseudo-monotone VIPs in Hilbert spaceshttps://zbmath.org/1491.650412022-09-13T20:28:31.338867Z"Thong, Duong Viet"https://zbmath.org/authors/?q=ai:duong-viet-thong."Dong, Qiao-Li"https://zbmath.org/authors/?q=ai:dong.qiaoli"Liu, Lu-Lu"https://zbmath.org/authors/?q=ai:liu.lulu"Triet, Nguyen Anh"https://zbmath.org/authors/?q=ai:nguyen-anh-triet."Lan, Nguyen Phuong"https://zbmath.org/authors/?q=ai:lan.nguyen-phuongSummary: In this work, we propose two new iterative schemes for finding an element of the set of solutions of a \textit{pseudo-monotone}, Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak and strong convergence theorems are presented. The advantage of the proposed algorithms is that they do not require prior knowledge of the Lipschitz constant of the variational inequality mapping and only compute one projection onto a feasible set per iteration as well as without using the sequentially weakly continuity of the associated mapping. Under additional strong pseudo-monotonicity and Lipschitz continuity assumptions, we obtain also an \(R\)-linear convergence rate of the proposed algorithm. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms.Ritz-like values in steplength selections for stochastic gradient methodshttps://zbmath.org/1491.650422022-09-13T20:28:31.338867Z"Franchini, Giorgia"https://zbmath.org/authors/?q=ai:franchini.giorgia"Ruggiero, Valeria"https://zbmath.org/authors/?q=ai:ruggiero.valeria"Zanni, Luca"https://zbmath.org/authors/?q=ai:zanni.lucaSummary: The steplength selection is a crucial issue for the effectiveness of the stochastic gradient methods for large-scale optimization problems arising in machine learning. In a recent paper, \textit{R. Bollapragada} et al. [SIAM J. Optim. 28, No. 4, 3312--3343 (2018; Zbl 1461.65131)] propose to include an adaptive subsampling strategy into a stochastic gradient scheme, with the aim to assure the descent feature in expectation of the stochastic gradient directions. In this approach, theoretical convergence properties are preserved under the assumption that the positive steplength satisfies at any iteration a suitable bound depending on the inverse of the Lipschitz constant of the objective function gradient. In this paper, we propose to tailor for the stochastic gradient scheme the steplength selection adopted in the full-gradient method knows as limited memory steepest descent method. This strategy, based on the Ritz-like values of a suitable matrix, enables to give a local estimate of the inverse of the local Lipschitz parameter, without introducing line search techniques, while the possible increase in the size of the subsample used to compute the stochastic gradient enables to control the variance of this direction. An extensive numerical experimentation highlights that the new rule makes the tuning of the parameters less expensive than the trial procedure for the efficient selection of a constant step in standard and mini-batch stochastic gradient methods.Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems on Riemannian manifoldshttps://zbmath.org/1491.650432022-09-13T20:28:31.338867Z"Li, Xiaobo"https://zbmath.org/authors/?q=ai:li.xiaobo"Krishan Lal, Manish"https://zbmath.org/authors/?q=ai:krishan-lal.manishSummary: We propose two nonmonotone projected gradient methods for multiobjective optimization problems on Riemannian manifolds. The global convergence to the Pareto critical point is proved under some reasonable conditions. Moreover, under some convexity assumptions, we also establish the full convergence to the Pareto critical point produced by the proposed algorithm on Riemannian manifolds with nonnegative curvature.Two limited-memory optimization methods with minimum violation of the previous secant conditionshttps://zbmath.org/1491.650442022-09-13T20:28:31.338867Z"Vlček, Jan"https://zbmath.org/authors/?q=ai:vlcek.jan"Lukšan, Ladislav"https://zbmath.org/authors/?q=ai:luksan.ladislavSummary: Limited-memory variable metric methods based on the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) update are widely used for large scale optimization. The block version of this update, derived for general objective functions in [\textit{J. Vlček} and \textit{L. Lukšan}, Numer. Algorithms 80, No. 3, 957--987 (2019; Zbl 1440.90076)], satisfies the secant conditions with all used difference vectors and for quadratic objective functions gives the best improvement of convergence in some sense, but the corresponding direction vectors are not descent directions generally. To guarantee the descent property of direction vectors and simultaneously violate the secant conditions as little as possible in some sense, two methods based on the block BFGS update are proposed. They can be advantageously used together with methods based on vector corrections for conjugacy. Here we combine two types of these corrections to satisfy the secant conditions with both the corrected and uncorrected (original) latest difference vectors. Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new methods.Bregman Itoh-Abe methods for sparse optimisationhttps://zbmath.org/1491.650452022-09-13T20:28:31.338867Z"Benning, Martin"https://zbmath.org/authors/?q=ai:benning.martin"Riis, Erlend Skaldehaug"https://zbmath.org/authors/?q=ai:riis.erlend-skaldehaug"Schönlieb, Carola-Bibiane"https://zbmath.org/authors/?q=ai:schonlieb.carola-bibianeSummary: In this paper we propose optimisation methods for variational regularisation problems based on discretising the inverse scale space flow with discrete gradient methods. Inverse scale space flow generalises gradient flows by incorporating a generalised Bregman distance as the underlying metric. Its discrete-time counterparts, Bregman iterations and linearised Bregman iterations are popular regularisation schemes for inverse problems that incorporate a priori information without loss of contrast. Discrete gradient methods are tools from geometric numerical integration for preserving energy dissipation of dissipative differential systems. The resultant Bregman discrete gradient methods are unconditionally dissipative and achieve rapid convergence rates by exploiting structures of the problem such as sparsity. Building on previous work on discrete gradients for non-smooth, non-convex optimisation, we prove convergence guarantees for these methods in a Clarke subdifferential framework. Numerical results for convex and non-convex examples are presented.Approximating the total variation with finite differences or finite elementshttps://zbmath.org/1491.650462022-09-13T20:28:31.338867Z"Chambolle, Antonin"https://zbmath.org/authors/?q=ai:chambolle.antonin"Pock, Thomas"https://zbmath.org/authors/?q=ai:pock.thomasSummary: We present and compare various types of discretizations which have been proposed to approximate the total variation (mostly, of a gray-level image in two dimensions). We discuss the properties of finite differences and finite elements based approach and compare their merits, in particular in terms of error estimates and quality of the reconstruction.
For the entire collection see [Zbl 1458.35003].The use of grossone in elastic net regularization and sparse support vector machineshttps://zbmath.org/1491.650472022-09-13T20:28:31.338867Z"De Leone, Renato"https://zbmath.org/authors/?q=ai:de-leone.renato"Egidi, Nadaniela"https://zbmath.org/authors/?q=ai:egidi.nadaniela"Fatone, Lorella"https://zbmath.org/authors/?q=ai:fatone.lorellaSummary: New algorithms for the numerical solution of optimization problems involving the \(l_0\) pseudo-norm are proposed. They are designed to use a recently proposed computational methodology that is able to deal numerically with finite, infinite and infinitesimal numbers. This new methodology introduces an infinite unit of measure expressed by the numeral \textcircled{1} (\textit{grossone}) and indicating the number of elements of the set \(\mathbb{N}\), of natural numbers. We show how the numerical system built upon \textcircled{1} and the proposed approximation of the \(l_0\) pseudo-norm in terms of \textcircled{1} can be successfully used in the solution of elastic net regularization problems and sparse support vector machines classification problems.
Editorial remark: For more information on the notion of grossone, introduced by \textit{Y. D. Sergeyev}, see [Arithmetic of infinity. Cosenza: Edizioni Orizzonti Meridionali (2003; Zbl 1076.03048); EMS Surv. Math. Sci. 4, No. 2, 219--320 (2017; Zbl 1390.03048)]; see also [\textit{A. E. Gutman} and \textit{S. S. Kutateladze}, Sib. Mat. Zh. 49, No. 5, 1054--1076 (2008; Zbl 1224.03045); translation in Sib. Math. J. 49, No. 5, 835--841 (2008)].Curve based approximation of measures on manifolds by discrepancy minimizationhttps://zbmath.org/1491.650482022-09-13T20:28:31.338867Z"Ehler, Martin"https://zbmath.org/authors/?q=ai:ehler.martin"Gräf, Manuel"https://zbmath.org/authors/?q=ai:graf.manuel"Neumayer, Sebastian"https://zbmath.org/authors/?q=ai:neumayer.sebastian"Steidl, Gabriele"https://zbmath.org/authors/?q=ai:steidl.gabrieleSummary: The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve's length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the \(2\)- and \(3\)-dimensional torus, the \(2\)-sphere, the rotation group on \(\mathbb{R}^3\) and the Grassmannian of all \(2\)-dimensional linear subspaces of \(\mathbb{R}^4\). Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removinghttps://zbmath.org/1491.650492022-09-13T20:28:31.338867Z"Nachaoui, Mourad"https://zbmath.org/authors/?q=ai:nachaoui.mourad"Afraites, Lekbir"https://zbmath.org/authors/?q=ai:afraites.lekbir"Hadri, Aissam"https://zbmath.org/authors/?q=ai:hadri.aissam"Laghrib, Amine"https://zbmath.org/authors/?q=ai:laghrib.amineSummary: This paper introduce a novel optimization procedure to reduce mixture of Gaussian and impulse noise from images. This technique exploits a non-convex PDE-constrained characterized by a fractional-order operator. The used non-convex term facilitated the impulse component approximation controlled by a spatial parameter \(\gamma\). A non-convex and non-smooth bi-level optimization framework with a modified projected gradient algorithm is then proposed in order to learn the parameter \(\gamma\). Denoising tests confirm that the non-convex term and learned parameter \(\gamma\) lead in general to an improved reconstruction when compared to results of convex norm and manual parameter \(\lambda\) choice.Metrically regular mapping and its utilization to convergence analysis of a restricted inexact Newton-type methodhttps://zbmath.org/1491.650502022-09-13T20:28:31.338867Z"Rashid, Mohammed Harunor"https://zbmath.org/authors/?q=ai:rashid.mohammed-harunorSummary: In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation \(0\in f(x)+F(x)\), where \(X\) and \(Y\) are Banach spaces, \(f:X\to Y\) is a Fréchet differentiable function and \(F\colon X\rightrightarrows Y\) is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of \(f\). Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of \(f\) is continuous and Lipschitz continuous as well as \(f+F\) is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics modelhttps://zbmath.org/1491.650512022-09-13T20:28:31.338867Z"Shutyaev, Victor"https://zbmath.org/authors/?q=ai:shutyaev.victor-petrovich"Parmuzin, Eugene"https://zbmath.org/authors/?q=ai:parmuzin.eugene-i"Gejadze, Igor"https://zbmath.org/authors/?q=ai:gejadze.igor-yuSummary: The problem of stability and sensitivity of functionals of the optimal solution of the variational data assimilation of sea surface temperature for the model of sea thermodynamics is considered. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. Computing the gradient of the response function reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. Stability analysis of the response function with respect to uncertainties of input data is given. Numerical examples are presented for the Black and Azov seas thermodynamics model.An inertial extragradient method for iteratively solving equilibrium problems in real Hilbert spaceshttps://zbmath.org/1491.650522022-09-13T20:28:31.338867Z"Rehman, Habib ur"https://zbmath.org/authors/?q=ai:rehman.habib-ur"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Shutaywi, Meshal"https://zbmath.org/authors/?q=ai:shutaywi.meshal"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapol"Wairojjana, Nopparat"https://zbmath.org/authors/?q=ai:wairojjana.nopparatSummary: In this article, we present an inertial subgradient extragradient-type method that uses a non-monotone step size rule to find a numerical solution to equilibrium problems in real Hilbert spaces. The presented iterative scheme is based on an extragradient subgradient method and an inertial-type scheme. In fact, the proposed iterative scheme is effective in terms of performance, and the key advantage derives directly from the use of the variable step size rule, which is revised by each iteration on the basis of the Lipschitz-type constants as well as certain prior iterations. We obtain a weak convergence theorem for a new method by using mild conditions on a bifunction. Applications of the main results are given to solve various nonlinear problems. Several numerical findings are given in order to illustrate the numerical behaviour of the proposed method and to compare it to others.Fractional Bernstein operational matrices for solving integro-differential equations involved by Caputo fractional derivativehttps://zbmath.org/1491.650532022-09-13T20:28:31.338867Z"Alshbool, M. H. T."https://zbmath.org/authors/?q=ai:alshbool.m-h-t"Mohammad, Mutaz"https://zbmath.org/authors/?q=ai:mohammad.mutaz"Isik, Osman"https://zbmath.org/authors/?q=ai:isik.osman-rasit"Hashim, Ishak"https://zbmath.org/authors/?q=ai:hashim.ishakSummary: The present work is devoted to developing two numerical techniques based on fractional Bernstein polynomials, namely fractional Bernstein operational matrix method, to numerically solving a class of fractional integro-differential equations (FIDEs). The first scheme is introduced based on the idea of operational matrices generated using integration, whereas the second one is based on operational matrices of differentiation using the collocation technique. We apply the Riemann-Liouville and fractional derivative in Caputo's sense on Bernstein polynomials, to obtain the approximate solutions of the proposed FIDEs. We also provide the residual correction procedure for both methods to estimate the absolute errors. Some results of the perturbation and stability analysis of the methods are theoretically and practically presented. We demonstrate the applicability and accuracy of the proposed schemes by a series of numerical examples. The numerical simulation exactly meets the exact solution and reaches almost zero absolute error whenever the exact solution is a polynomial. We compare the algorithms with some known analytic and semi-analytic methods. As a result, our algorithm based on the Bernstein series solution methods yield better results and show outstanding and optimal performance with high accuracy orders compared with those obtained from the optimal homotopy asymptotic method, standard and perturbed least squares method, CAS and Legendre wavelets method, and fractional Euler wavelet method.An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equationshttps://zbmath.org/1491.650542022-09-13T20:28:31.338867Z"Behera, S."https://zbmath.org/authors/?q=ai:behera.srikanta"Saha Ray, S."https://zbmath.org/authors/?q=ai:saha-ray.santanuSummary: The main purpose of this article is to solve the pantograph Volterra delay integro-differential equation of fractional order. A numerical operational matrix approach based on Euler wavelets is proposed. For the proposed scheme, the fractional integral operational matrix is constructed. Then the pantograph Volterra delay integro-differential equations are reduced to algebraic equations by using the fractional integral operational matrix. Several theorems are presented to establish the convergence and error analysis of the proposed method. To show the accuracy of the proposed technique, the numerical convergence rate has been shown. Additionally, some numerical problems are solved to justify the applicability and validity of the presented technique. Also, the numerical results have been documented graphically to describe the effectiveness of the approach. Furthermore, comparing numerical results with those obtained by known methods shows that the approach scheme is more efficient and accurate.Eigenvalues and delay differential equations: periodic coefficients, impulses and rigorous numericshttps://zbmath.org/1491.650552022-09-13T20:28:31.338867Z"Church, Kevin E. M."https://zbmath.org/authors/?q=ai:church.kevin-e-mSummary: We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We consider systems with a single delay where the period is at most equal to the delay, and the latter two are commensurate. We first represent the monodromy operator (period map) as an operator acting on a product of sequence spaces that represent the Chebyshev coefficients of the state-space vectors. Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation error in addition to some other technical operator norms, this leads to the method being suitable to computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs on two example problems. We also test our discretization scheme in floating point arithmetic on a gamut of randomly-generated high-dimensional examples with both periodic and constant coefficients to inspect the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check for periodic systems) for increasing numbers of Chebyshev modes.Higher order numerical schemes for the solution of fractional delay differential equationshttps://zbmath.org/1491.650562022-09-13T20:28:31.338867Z"Gande, Naga Raju"https://zbmath.org/authors/?q=ai:gande.naga-raju"Madduri, H."https://zbmath.org/authors/?q=ai:madduri.harshitaSummary: Caputo and Riemann-Liouville (R-L) are the most commonly used fractional derivative operators in the field of fractional calculus. We present here two higher-order schemes, based on R-L and Caputo definitions for the solution of fractional delay differential equations (FDDEs). For the R-L fractional derivative, interpolation-based approximation and the finite difference based approximation for Caputo fractional (C-F) derivative are employed to obtain the higher-order schemes i.e., \(\mathcal{O}(h^{3-\alpha})\), \((\alpha\in(0,1)\), \textit{the fractional order}) for FDDEs. For the proposed schemes the stability and error estimates are presented. The noted distinct feature is that the finite difference based approximation takes almost 50\% less computation time than the interpolation-based approximation. Varied examples have been worked out to verify the efficacy of the schemes, including non-linear problems such as, the chaotic behavior occurring in the fractional order Ikeda equation, the fractional order version of the population dynamics of Lemmings and the dynamical model for sea surface temperature.Analysis of an adaptive collocation solution for retarded and neutral delay systemshttps://zbmath.org/1491.650572022-09-13T20:28:31.338867Z"Maleki, Mohammad"https://zbmath.org/authors/?q=ai:maleki.mohammad-reza"Davari, Ali"https://zbmath.org/authors/?q=ai:davari.aliSummary: This paper introduces an adaptive collocation method to solve retarded and neutral delay differential equations (RDDEs and NDDEs) with constant or time-dependent delays. The delays are allowed to be small or become vanishing during the integration. We determine the convergence properties of the proposed method for neutral equations with solutions in appropriate Sobolev spaces. It is shown that the proposed scheme enjoys the spectral accuracy. Numerical results show that the proposed method can be implemented in an efficient and accurate manner for a wide range of RDDE and NDDE model problems.A-stable, explicit method for solving stiff problems in science and engineeringhttps://zbmath.org/1491.650582022-09-13T20:28:31.338867Z"Chang, Shuenn-Yih"https://zbmath.org/authors/?q=ai:chang.shuenn-yihSummary: Problem-dependent methods is a novel type of numerical methods in the solution of ordinary differential equations since their coefficients can be functions of initial properties to define the problem for analysis. A one-step, explicit problem-dependent method is shown to be A-stable and second order accurate. Numerical properties of this method are almost the same as the trapezoidal method except for the explicitness of each step. It is affirmed by means of solving a variety of nonlinear first order stiff problems that it can have almost the same performance as that of the trapezoidal method. However, the most important improvement of this method in contrast to the trapezoidal method is that it can solve such nonlinear problems non-iteratively. Numerical tests reveal that the CPU demand for the method is as small as 1.9, 1.3, 1.0 and 0.51\% of that consumed by the trapezoidal method for solving a set of 250, 500, 1000 and 2000 nonlinear ODEs of first order. Hence, it is much cheaper than the trapezoidal method in computing when solving stiff systems of nonlinear problems.Global error estimation for explicit general linear methodshttps://zbmath.org/1491.650592022-09-13T20:28:31.338867Z"Abdi, Ali"https://zbmath.org/authors/?q=ai:abdi.ali"Hojjati, Gholamreza"https://zbmath.org/authors/?q=ai:hojjati.gholamreza"Izzo, Giuseppe"https://zbmath.org/authors/?q=ai:izzo.giuseppe"Jackiewicz, Zdzislaw"https://zbmath.org/authors/?q=ai:jackiewicz.zdzislawSummary: We describe an approach to global error estimation for explicit general linear methods. This approach is based on computation of two numerical solutions by pairs of general linear methods of the same order and stage order and with proportional global error functions. The results of numerical experiments indicate that this approach is very accurate in constant and variable stepsize environments.Global error estimation for explicit second derivative general linear methodshttps://zbmath.org/1491.650602022-09-13T20:28:31.338867Z"Abdi, Ali"https://zbmath.org/authors/?q=ai:abdi.ali"Hojjati, Gholamreza"https://zbmath.org/authors/?q=ai:hojjati.gholamreza"Izzo, Giuseppe"https://zbmath.org/authors/?q=ai:izzo.giuseppe"Jackiewicz, Zdzislaw"https://zbmath.org/authors/?q=ai:jackiewicz.zdzislawSummary: In this paper, we describe an approach to estimate the global error for explicit second derivative general linear methods based on the approach which has been already used for global error estimation of explicit general linear methods. In this approach, to estimate the global error, we use the numerical solutions of pairs of second derivative general linear methods with the same order and stage order that are constructed such that their global error functions are proportional. Numerical experiments demonstrate the excellent agreement of the global error estimation with the exact one in both constant and variable stepsize environments.Approximating the solution of the differential equations with fractional operatorshttps://zbmath.org/1491.650612022-09-13T20:28:31.338867Z"Ahmed, Rana Talha"https://zbmath.org/authors/?q=ai:ahmed.rana-talha"Sohail, Ayesha"https://zbmath.org/authors/?q=ai:sohail.ayeshaSummary: We described a brief history of fractional calculus from sixteenth century to twentieth century. Basic functions like gamma function and Mittag-Leffler function are defined which help to understand fractional calculus. Popular fractional integral and derivative operators are also defined. Fractional order ordinary and partial differential equations are also introduced. In this research, we study and proposed two basic methods to solve fractional order ordinary differential equations like semi-analytical method which is followed by Laplace transform and a numerical method which is followed by backward (implicit) Euler's method. To support these methods we also solved a few examples for better understanding.Arbitrary high-order methods for one-sided direct event location in discontinuous differential problems with nonlinear event functionhttps://zbmath.org/1491.650622022-09-13T20:28:31.338867Z"Amodio, Pierluigi"https://zbmath.org/authors/?q=ai:amodio.pierluigi"Brugnano, Luigi"https://zbmath.org/authors/?q=ai:brugnano.luigi"Iavernaro, Felice"https://zbmath.org/authors/?q=ai:iavernaro.feliceSummary: In this paper we are concerned with numerical methods for the one-sided event location in discontinuous differential problems, whose event function is nonlinear (in particular, of polynomial type). The original problem is transformed into an equivalent Poisson problem, which is effectively solved by suitably adapting a recently devised class of energy-conserving methods for Poisson systems. The actual implementation of the methods is fully discussed, with a particular emphasis to the problem at hand. Some numerical tests are reported, to assess the theoretical findings.An energy preserving discretization method for the thermodynamic Kuramoto model and collective behaviorshttps://zbmath.org/1491.650632022-09-13T20:28:31.338867Z"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Shim, Woojoo"https://zbmath.org/authors/?q=ai:shim.woojoo"Yoon, Jaeyoung"https://zbmath.org/authors/?q=ai:yoon.jaeyoungSummary: We provide an energy preserving discretization method for the thermodynamic Kuramoto (TK) model on a lattice and investigate its emergent dynamics, and show a smooth transition from the proposed discrete model to the corresponding continuous model. The thermodynamic Kuramoto model describes the temporal evolution of the phase and temperature at each lattice point in a domain. To integrate the continuous model numerically, one needs to discretize the continuous model in a suitable way so that the resulting discrete model exhibits the same emergent features as the corresponding continuous model. The naive forward Euler discretization for phase-temperature configuration does not conserve a total energy, which causes inconsistency with the continuous model. Thus, we instead propose an implicit scheme which preserves energy and satisfies entropy principle, and provide several sufficient frameworks leading to the emergent collective behaviors and uniform-in-time smooth transition from the discrete model to the continuous model.Perturbed Runge-Kutta methods for mixed precision applicationshttps://zbmath.org/1491.650642022-09-13T20:28:31.338867Z"Grant, Zachary J."https://zbmath.org/authors/?q=ai:grant.zachary-jSummary: In this work we consider a mixed precision approach to accelerate the implementation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-precision computation without adversely impacting the accuracy of the overall solution. In particular, a properly designed Runge-Kutta method will damp out the errors committed in the initial stages. This is of particular interest when we consider implicit Runge-Kutta methods. In such cases, the implicit computation of the stage values can be considerably faster if the solution can be of lower precision (or, equivalently, have a lower tolerance). We provide a general theoretical additive framework for designing mixed precision Runge-Kutta methods, and use this framework to derive order conditions for such methods. Next, we show how using this approach allows us to leverage low precision computation of the implicit solver while retaining high precision in the overall method. We present the behavior of some mixed-precision implicit Runge-Kutta methods through numerical studies, and demonstrate how the numerical results match with the theoretical framework. This novel mixed-precision implicit Runge-Kutta framework opens the door to the design of many such methods.Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearitieshttps://zbmath.org/1491.650652022-09-13T20:28:31.338867Z"Horsin, Thierry"https://zbmath.org/authors/?q=ai:horsin.thierry"Jendoubi, Mohamed Ali"https://zbmath.org/authors/?q=ai:jendoubi.mohamed-aliSummary: In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.Approximation scheme for handling coupled systems of differential equations within reproducing kernel methodhttps://zbmath.org/1491.650662022-09-13T20:28:31.338867Z"Ateiwi, Ali"https://zbmath.org/authors/?q=ai:ateiwi.ali-mahmud"Édamat, Ayed Al"https://zbmath.org/authors/?q=ai:al-edamat.ayed"Freihat, Asad"https://zbmath.org/authors/?q=ai:freihat.asad-a"Komashynska, Iryna"https://zbmath.org/authors/?q=ai:komashynska.iryna-volodymyrivnaSummary: This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of systems of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In
this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.Numerical computation of the coefficients in exponential fittinghttps://zbmath.org/1491.650672022-09-13T20:28:31.338867Z"Ixaru, L. Gr."https://zbmath.org/authors/?q=ai:ixaru.liviu-grTwo classes of Numerov methods based on exponential fitting are considered to solve the Schrödinger equation. The first class of methods is defined by methods depending on three parameters that includes the well-known methods \(S_m\), \(m=0,1,2,3\), characterized by known analytical expressions for the coefficients. These relations are established in the literature under a convenient condition: the maximum number of parameters that can be different is only two. The second class is defined by methods depending on five parameters.
For general Numerov methods based on exponential fitting dependent on three or five parameters, linear systems are established for the coefficients and no closed formulas are known. This problem is solved in this paper using subroutines accessible in the literature. The analysis of the accuracy of the new methods is presented.
Reviewer: José Augusto Ferreira (Coimbra)Galerkin finite element method for nonlinear fractional differential equationshttps://zbmath.org/1491.650682022-09-13T20:28:31.338867Z"Nedaiasl, Khadijeh"https://zbmath.org/authors/?q=ai:nedaiasl.khadijeh"Dehbozorgi, Raziyeh"https://zbmath.org/authors/?q=ai:dehbozorgi.raziyehThe paper studies some analytical properties of a boundary value problem with a fractional ordinary differential equation of Riemann-Liouville or Caputo type whose order is in \((1,2)\), associated with homogeneous Dirichlet boundary conditions and a specific type of nonlinearities. For the approximate solution of such problems under certain conditions on the given data, a finite element method is described and investigated.
Reviewer: Kai Diethelm (Schweinfurt)The approximate solution of the third order boundary value problem with an internal boundary condition using a hybrid finite difference methodhttps://zbmath.org/1491.650692022-09-13T20:28:31.338867Z"Pandey, Pramod Kumar"https://zbmath.org/authors/?q=ai:kumar-pandey.pramod"Mishra, Basant Kumar"https://zbmath.org/authors/?q=ai:mishra.basant-kumarSummary: In the present article, we considered and proposed a numerical technique for the solution of differential equations of order three. The boundary conditions are prescribed on the two points on the boundary and an interior point in the domain of the reference differential equation. Our technique based on method of finite difference approximations and a system of linear equations obtained after discretization of the continuous problem. The development and the convergence of the proposed method discussed in detail. Additionally, in the numerical experiment considered both linear and nonlinear model problems. The computational efficiency and quadratic order of the convergence of the proposed HFD method tested by experiment. The proposed HFD technique is accurate and precise, it may be seen from tabulated numerical outcomes.An efficient numerical approach for solving a general class of nonlinear singular boundary value problemshttps://zbmath.org/1491.650702022-09-13T20:28:31.338867Z"Roul, Pradip"https://zbmath.org/authors/?q=ai:roul.pradip"Kumari, Trishna"https://zbmath.org/authors/?q=ai:kumari.trishna"Prasad Goura, V. M. K."https://zbmath.org/authors/?q=ai:prasad-goura.v-m-kSummary: This paper is concerned with the development of a collocation method based on the Bessel polynomials for numerical solution of a general class of nonlinear singular boundary value problems (SBVPs). Due to the existence of singularity at the point \(x=0\), we first modify the problem at the singular point. The proposed method is then developed for solving the resulting regular boundary value problem. To demonstrate the effectiveness and accuracy of the method, we apply it on several numerical examples. The numerical results obtained confirm that the present method has an advantage in terms of numerical accuracy over the uniform mesh cubic B-spline collocation (UCS) method [\textit{P. Roul} and \textit{V. M. K. Prasad Goura}, Appl. Math. Comput. 341, 428--450 (2019; Zbl 1429.65164)], non-standard finite difference (NSFD) method [\textit{A. K. Verma} and \textit{S. Kayenat}, J. Math. Chem. 56, No. 6, 1667--1706 (2018; Zbl 1448.65121)], three-point finite difference methods (FDMs) [\textit{R. K. Pandey} and \textit{A. K. Singh}, Int. J. Comput. Math. 80, No. 10, 1323--1331 (2003; Zbl 1047.65060); J. Comput. Appl. Math. 205, No. 1, 469--478 (2007; Zbl 1149.65065)] and the cubic B-spline collocation (CBSC) method [\textit{H. Çağlar} et al., Chaos Solitons Fractals 39, No. 3, 1232--1237 (2009; Zbl 1197.65107)].A uniform discretization for solving singularly perturbed convection-diffusion boundary value problemshttps://zbmath.org/1491.650712022-09-13T20:28:31.338867Z"Gunes, Baransel"https://zbmath.org/authors/?q=ai:gunes.baransel"Demirbas, Mutlu"https://zbmath.org/authors/?q=ai:demirbas.mutluSummary: In this paper, a discrete scheme is presented for solving singularly perturbed convection-diffusion equations. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Error estimates are carried out for both Bakhvalov \((B\)-mesh) and Shishkin-type \((S\)-mesh) meshes. Three numerical examples are solved to authenticate the theoretical findings.A new hybrid collocation method for solving nonlinear two-point boundary value problemshttps://zbmath.org/1491.650722022-09-13T20:28:31.338867Z"Delpasand, Razieh"https://zbmath.org/authors/?q=ai:delpasand.razieh"Hosseini, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:hosseini.mohammad-mehdi"Ghaini, Farid Mohammad Maalek"https://zbmath.org/authors/?q=ai:maalek-ghaini.farid-mohammadSummary: In this paper, numerical solution of boundary value problems (BVPs) of nonlinear ordinary differential equations (ODEs) by the collocation method is considered. Of course, to avoid solving systems of nonlinear algebraic equations resulting from the method, residual function of the boundary value problem is considered and an unconstrained optimisation model is suggested. Particle swarm optimisation (PSO) algorithm is used for solving the unconstrained optimisation problem. In addition, convergence properties of the Chebyshev expansion are studied. The scheme is tested on some interesting examples and the obtained results demonstrate reliability and efficiency of the proposed hybrid method.WKB-based scheme with adaptive step size control for the Schrödinger equation in the highly oscillatory regimehttps://zbmath.org/1491.650732022-09-13T20:28:31.338867Z"Körner, Jannis"https://zbmath.org/authors/?q=ai:korner.jannis"Arnold, Anton"https://zbmath.org/authors/?q=ai:arnold.anton"Döpfner, Kirian"https://zbmath.org/authors/?q=ai:dopfner.kirianSummary: This paper is concerned with an efficient numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Being a hybrid, analytical-numerical approach it does not have to resolve each oscillation, in contrast to standard schemes for ODEs. We build upon the WKB-based (named after the physicists Wentzel, Kramers, Brillouin) marching method from [\textit{A. Arnold} et al., SIAM J. Numer. Anal. 49, No. 4, 1436--1460 (2011; Zbl 1230.65078)] and extend it in two ways: By comparing the \(\mathcal{O} ( h )\) and \(\mathcal{O} ( h^2 )\) methods from [\textit{A. Arnold} et al., SIAM J. Numer. Anal. 49, No. 4, 1436--1460 (2011; Zbl 1230.65078)] we design an adaptive step size controller for the WKB method. While this WKB method is very efficient in the highly oscillatory regime, it cannot be used close to turning points. Hence, we introduce for such regions an automated methods switching, choosing between the WKB method for the oscillatory region and a standard Runge-Kutta-Fehlberg 4(5) method in smooth regions. A similar approach was proposed recently in [\textit{W. J. Handley}, \textit{A. N. Lasenby} and \textit{M. P. Hobson}, ``The Runge-Kutta-Wentzel-Kramers-Brillouin method'', Preprint, \url{arXiv:1612.02288}] and [\textit{F. J. Agocs} et al., ``An efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems'', Preprint, \url{arXiv:1906.01421}], however, only for an \(\mathcal{O} ( h )\)-method. Hence, we compare our new strategy to their method on two examples (Airy function on the spatial interval \([ 0 , 1 0^8 ]\) with one turning point at \(x = 0\) and on a parabolic cylinder function having two turning points), and illustrate the advantages of the new approach w.r.t. accuracy and efficiency.Numerical implementation of Daftardar-Gejji and Jafari method to the quadratic Riccati equationhttps://zbmath.org/1491.650742022-09-13T20:28:31.338867Z"Batiha, Belal"https://zbmath.org/authors/?q=ai:batiha.belal"Ghanim, Firas"https://zbmath.org/authors/?q=ai:ghanim.firas(no abstract)Solving differential equations with artificial bee colony programminghttps://zbmath.org/1491.650752022-09-13T20:28:31.338867Z"Boudouaoui, Yassine"https://zbmath.org/authors/?q=ai:boudouaoui.yassine"Habbi, Hacene"https://zbmath.org/authors/?q=ai:habbi.hacene"Ozturk, Celal"https://zbmath.org/authors/?q=ai:ozturk.celal"Karaboga, Dervis"https://zbmath.org/authors/?q=ai:karaboga.dervisSummary: Relying on artificial bee colony programming (ABCP), we present in this paper, for the first time, a novel methodology for solving differential equations. The three-phase evolving process of ABCP is managed to apply on the issue of recovering the exact solution of differential equations through a well-posed problem. In fact, the original ABCP model which has been initially developed for symbolic regression cannot be used directly as differential problems might have multiple outputs. Moreover, the definition of fitness function is a critical problem-dependent issue for model design. In this sense, a problem-specific ABCP algorithm is worked out in the present contribution. With the proposed algorithm, solution with multiple outputs can evolve under a multiple-tree framework toward the exact solution. For fitness function evaluation, different forms are derived for ordinary and partial differential equations by performing experiments with multiple runs. Results on several differential equations are reported and compared to other advanced methods to assess the feasibility and the potential of the proposed method. A computational performance evaluation is provided for the considered examples and completed with an additional study on the impact of key control parameters.Relationship between time-instant number and precision of ZeaD formulas with proofshttps://zbmath.org/1491.650762022-09-13T20:28:31.338867Z"Yang, Min"https://zbmath.org/authors/?q=ai:yang.min"Zhang, Yunong"https://zbmath.org/authors/?q=ai:zhang.yunong"Hu, Haifeng"https://zbmath.org/authors/?q=ai:hu.haifengFor time-varying numerical matrix problems, the paper studies the coefficients \(a_i\) of look-ahead and convergent discretization formulas \(\dot x_k = a_{m-1} x_{k+1}/\tau + \dots + a_0 x_{k-m+2}/\tau + O(\tau^p)\) that are used in Zhang Neural Networks (ZNN). The standard approach uses \(m\) Taylor expansions of \(x_{k+1}\) through \(x_{k-m+2}\) around \(x_k\) for a constant sampling gap \(\tau = x_j - x_{j-1}\). Forming the linear combination of these Taylor expansions with the coefficients \(a_{m-1}, \dots, a_0\) in turn then leads to a linear system for feasible \(a_j\) coefficients. If \(m > p+1\), then the linear system in \(a_j\) is solvable over the rational numbers and the relations between the coefficients can be exploited to reduce the solution set further. This method is applied to exhibit and study look-ahead and convergent finite difference formulas for various \(m\) and \(p\) values in light of their length \(m\) and their innate truncation error order \(O(\tau^p)\) on a time-varying quadratic optimization problem.
Reviewer: Frank Uhlig (Auburn)Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systemshttps://zbmath.org/1491.650772022-09-13T20:28:31.338867Z"Allwright, Amy"https://zbmath.org/authors/?q=ai:allwright.amy"Atangana, Abdon"https://zbmath.org/authors/?q=ai:atangana.abdonSummary: The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established.A new linearized fourth-order conservative compact difference scheme for the SRLW equationshttps://zbmath.org/1491.650782022-09-13T20:28:31.338867Z"He, Yuyu"https://zbmath.org/authors/?q=ai:he.yuyu"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.4|wang.xiaofeng|wang.xiaofeng.1|wang.xiaofeng.3|wang.xiaofeng.2"Zhong, Ruihua"https://zbmath.org/authors/?q=ai:zhong.ruihuaSummary: In this paper, a novel three-point fourth-order compact operator is considered to construct new linearized conservative compact finite difference scheme for the symmetric regularized long wave (SRLW) equations based on the reduction order method with three-level linearized technique. The discrete conservative laws, boundedness and unique solvability are studied. The convergence order \(\mathcal{O}(\tau^2+h^4)\) in the \(L^{\infty}\)-norm and stability of the present compact scheme are proved by the discrete energy method. Numerical examples are given to support the theoretical analysis.Computer turbulence as a tunnelling effecthttps://zbmath.org/1491.650792022-09-13T20:28:31.338867Z"Sharkovsky, A. N."https://zbmath.org/authors/?q=ai:sharkovskyi.oleksandr-m"Romanenko, E. Yu."https://zbmath.org/authors/?q=ai:romanenko.elena-yu"Akbergenov, A. A."https://zbmath.org/authors/?q=ai:akbergenov.a-aIn this paper several specific features of computer turbulence is discussed. This phenomena occurs in problems of mathematical physics where the exact solutions have smooth dynamics, but their discrete analogs (numerical approximations) lead to the emergence, at a certain time, of the ``Brownian'' dynamics. This is usually considered as the result of incorrect calculations caused by the discretization of the original (continuous) problems. On the other hand these models may give a more adequate description of the material world then the continuous models, and in this case this phenomena is called computer turbulence. In particular, it can be understand as the phenomenon that occurs in discrete models of mathematical physics characterized by the emergence of Brownian-like dynamics at a certain time. In this paper several examples are given and discussed. The computer tunnelling effect is addressed.
Reviewer: Petr Sváček (Praha)Error estimation of the Besse relaxation scheme for a semilinear heat equationhttps://zbmath.org/1491.650802022-09-13T20:28:31.338867Z"Zouraris, Georgios E."https://zbmath.org/authors/?q=ai:zouraris.georgios-eSummary: The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse Relaxation Scheme in time [\textit{C. Besse}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 12, 1427--1432 (1998; Zbl 0911.65072)] with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete \(L_t^\infty (H_x^2)\)-norm at the time-nodes and in the discrete \(L_t^\infty (H_x^1)\)-norm at the intermediate time-nodes. It is the first time in the literature where the Besse Relaxation Scheme is applied and analysed in the context of parabolic equations.Non-overlapping Schwarz algorithms for the incompressible Navier-Stokes equations with DDFV discretizationshttps://zbmath.org/1491.650812022-09-13T20:28:31.338867Z"Goudon, Thierry"https://zbmath.org/authors/?q=ai:goudon.thierry"Krell, Stella"https://zbmath.org/authors/?q=ai:krell.stella"Lissoni, Giulia"https://zbmath.org/authors/?q=ai:lissoni.giuliaThe authors consider the numerical resolution of the unsteady incompressible Navier-Stokes problem. They first establish the well-posedness of DDFV (Discrete Duality Finite Volume) schemes on the whole spatial domain with general convection fluxes defined by \(B\)-schemes. Subsequently, they propose two non-overlapping DDFV Schwarz algorithms. DDFV discretizations are constructed with suitable transmission conditions. When using standard convection fluxes in the domain decomposition method, the iterative process converges to a system with modified fluxes at the interface. However, it is possible to modify the fluxes of the domain decomposition algorithm so that it converges to the reference scheme on the entire domain. Some numerical tests are presented to illustrate the behavior and the performances of the algorithms
Reviewer: Abdallah Bradji (Annaba)Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth modelhttps://zbmath.org/1491.650822022-09-13T20:28:31.338867Z"Remesan, Gopikrishnan Chirappurathu"https://zbmath.org/authors/?q=ai:remesan.gopikrishnan-chirappurathuThe author considers the finite volume approximation, on nonuniform Cartesian grids, of the nonlinear scalar conservation law \(\partial_t \alpha +\operatorname{div}(u f(\alpha )) = 0\) in two and three spatial dimensions with an initial data of bounded variation. A uniform estimate on total variation of discrete solutions is proved. The standard assumption which states that the advecting velocity vector is divergence free is relaxed. Since the underlying meshes are nonuniform Cartesian, it is possible to adaptively refine the mesh on regions where the solution is expected to have sharp fronts. A uniform BV estimate is also obtained for finite volume approximations of conservation laws that has a fully nonlinear flux on nonuniform Cartesian grids. Some numerical tests are presented to support the theoretical results.
Reviewer: Abdallah Bradji (Annaba)Variable piecewise interpolation solution of the transport equationhttps://zbmath.org/1491.650832022-09-13T20:28:31.338867Z"Romm, Ya. E."https://zbmath.org/authors/?q=ai:romm.ya-e"Dzhanunts, G. A."https://zbmath.org/authors/?q=ai:dzhanunts.g-aSummary: In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments are performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed.A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equationshttps://zbmath.org/1491.650842022-09-13T20:28:31.338867Z"Wu, Shuonan"https://zbmath.org/authors/?q=ai:wu.shuonan"Zhou, Zhi"https://zbmath.org/authors/?q=ai:zhou.zhiA fully discrete low-regularity integrator for the nonlinear Schrödinger equationhttps://zbmath.org/1491.650852022-09-13T20:28:31.338867Z"Ostermann, Alexander"https://zbmath.org/authors/?q=ai:ostermann.alexander"Yao, Fangyan"https://zbmath.org/authors/?q=ai:yao.fangyanSummary: For the solution of the one dimensional cubic nonlinear Schrödinger equation on the torus, we propose and analyze a fully discrete low-regularity integrator. The considered scheme is explicit. Its implementation relies on the fast Fourier transform with a complexity of \(\mathcal{O}(N\log N)\) operations per time step, where \(N\) denotes the degrees of freedom in the spatial discretization. We prove that the new scheme provides an \(\mathcal{O}(\tau^{\frac{3}{2}\gamma -\frac{1}{2}-\varepsilon}+N^{-\gamma})\) error bound in \(L^2\) for any initial data in \(H^\gamma\), \(\frac{1}{2}<\gamma \leq 1\), where \(\tau\) denotes the temporal step size. Numerical examples illustrate this convergence behavior.Unconditionally positivity preserving and energy dissipative schemes for Poisson-Nernst-Planck equationshttps://zbmath.org/1491.650862022-09-13T20:28:31.338867Z"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jie"Xu, Jie"https://zbmath.org/authors/?q=ai:xu.jieSummary: We develop a set of numerical schemes for the Poisson-Nernst-Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results are presented to validate these properties. Moreover, numerical results indicate that the second-order scheme is also energy dissipative, and both the first- and the second-order scheme preserves the maximum principle for cases where the equation satisfies the maximum principle.A noniterative domain decomposition method for the forward-backward heat equationhttps://zbmath.org/1491.650872022-09-13T20:28:31.338867Z"Banei, S."https://zbmath.org/authors/?q=ai:banei.siamak"Shanazari, K."https://zbmath.org/authors/?q=ai:shanazari.kamalSummary: A nonoverlapping domain decomposition technique applied to a finite difference method is presented for the numerical solution of the forward backward heat equation in the case of one-dimension. While the previous attempts in dealing with this problem have been based on an iterative domain decomposition scheme, the current work avoids iterations. Also a physical matching condition is suggested to avoid difficulties caused by the interface boundary nodes. Furthermore, we obtain a square system of equations. In addition, the convergence and stability of the proposed method are investigated. Some numerical experiments are given to show the effectiveness of the proposed method.Parallel tridiagonal matrix inversion with a hybrid multigrid-Thomas algorithm methodhttps://zbmath.org/1491.650882022-09-13T20:28:31.338867Z"Parker, J. T."https://zbmath.org/authors/?q=ai:parker.jason-t"Hill, P. A."https://zbmath.org/authors/?q=ai:hill.p-a"Dickinson, D."https://zbmath.org/authors/?q=ai:dickinson.detta|dickinson.david-l"Dudson, B. D."https://zbmath.org/authors/?q=ai:dudson.b-dSummary: Tridiagonal matrix inversion is an important operation with many applications. It arises frequently in solving discretized one-dimensional elliptic partial differential equations, and forms the basis for many algorithms for block tridiagonal matrix inversion for discretized PDEs in higher-dimensions. In such systems, this operation is often the scaling bottleneck in parallel computation. In this paper, we derive a hybrid multigrid-Thomas algorithm designed to efficiently invert tridiagonal matrix equations in a highly-scalable fashion in the context of time evolving partial differential equation systems. We decompose the domain between processors, using multigrid to solve on a grid consisting of the boundary points of each processor's local domain. We then reconstruct the solution on each processor using a direct solve with the Thomas algorithm. This algorithm has the same theoretical optimal scaling as cyclic reduction and recursive doubling. We use our algorithm to solve Poisson's equation as part of the spatial discretization of a time-evolving PDE system. Our algorithm is faster than cyclic reduction per inversion and retains good scaling efficiency to twice as many cores.Stability and error estimates for non-linear Cahn-Hilliard-type equations on evolving surfaceshttps://zbmath.org/1491.650892022-09-13T20:28:31.338867Z"Beschle, Cedric Aaron"https://zbmath.org/authors/?q=ai:beschle.cedric-aaron"Kovács, Balázs"https://zbmath.org/authors/?q=ai:kovacs.balazsSummary: In this paper, we consider a non-linear fourth-order evolution equation of Cahn-Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix-vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.Semi-discrete finite-element approximation of nonlocal hyperbolic problemhttps://zbmath.org/1491.650902022-09-13T20:28:31.338867Z"Chaudhary, Sudhakar"https://zbmath.org/authors/?q=ai:chaudhary.sudhakar"Srivastava, Vimal"https://zbmath.org/authors/?q=ai:srivastava.vimalSummary: In this paper, we investigate a semi-discrete finite-element approximation of nonlocal hyperbolic problem. A priori error estimate for the semi-discrete scheme is derived. A fully discrete scheme based on backward difference method is constructed. We discuss the existence-uniqueness of the solution for fully discrete problem. In order to linearize the nonlinear fully discrete problem, we use Newton's method. Numerical results based on the usual finite-element method are provided to confirm the theoretical estimate.Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn-Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosityhttps://zbmath.org/1491.650912022-09-13T20:28:31.338867Z"Chen, Chuanjun"https://zbmath.org/authors/?q=ai:chen.chuanjun"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: We construct a fully-discrete finite element numerical scheme for the Cahn-Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier-Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn-Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh-Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noisehttps://zbmath.org/1491.650922022-09-13T20:28:31.338867Z"Feng, Xiaobing"https://zbmath.org/authors/?q=ai:feng.xiaobing"Qiu, Hailong"https://zbmath.org/authors/?q=ai:qiu.hailongSummary: This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler-Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite element methods.A space-time Trefftz discontinuous Galerkin method for the linear Schrödinger equationhttps://zbmath.org/1491.650932022-09-13T20:28:31.338867Z"Gómez, Sergio"https://zbmath.org/authors/?q=ai:gomez.sergio-alejandro"Moiola, Andrea"https://zbmath.org/authors/?q=ai:moiola.andreaThis paper proposes and analyzes a space-time Trefftz discontinuous Galerkin method for the Schrödinger equation with piecewise-constant potential. The main feature of Trefftzmethods is that they seek approximations in spaces spanned by local solutions ofthe PDE considered. This typically requires nonpolynomial basis functions. Trefftz scheme allows for much faster convergence in terms of degrees of freedom than classical polynomial DG schemes.This approach to the Trefftz approximation theory is completelydifferent from that used for the Helmholtz equation. The well-posedness and quasi-optimality of theTrefftz-DG approximation for arbitrary dimensions and discrete Trefftz subspaces are proved.The error analysis for discrete subspaces spanned by complexexponentials satisfying the Schrödinger equation is presented. Some numerical experiments validatethe theoretical results presented.
Reviewer: Yan Xu (Hefei)Numerical analysis for Maxwell obstacle problems in electric shieldinghttps://zbmath.org/1491.650942022-09-13T20:28:31.338867Z"Hensel, Maurice"https://zbmath.org/authors/?q=ai:hensel.maurice"Yousept, Irwin"https://zbmath.org/authors/?q=ai:yousept.irwinThis paper discusses a finite element method for a Maxwell obstacle problem in electric shielding. The approach relies on the leapfrog time-stepping and the Nedelec edge elements in which no additional nonlinear solver is required for the computation of the discrete evolutionary variational inequality of Ampere-Maxwell type. \(L^1\) and \(L^2\) stability are discussed and several numerical experiments are included.
Reviewer: Marius Ghergu (Dublin)Local transparent boundary conditions for wave propagation in fractal trees. I: Method and numerical implementationhttps://zbmath.org/1491.650952022-09-13T20:28:31.338867Z"Joly, Patrick"https://zbmath.org/authors/?q=ai:joly.patrick"Kachanovska, Maryna"https://zbmath.org/authors/?q=ai:kachanovska.marynaA nonsymmetric approach and a quasi-optimal and robust discretization for the Biot's modelhttps://zbmath.org/1491.650962022-09-13T20:28:31.338867Z"Khan, Arbaz"https://zbmath.org/authors/?q=ai:khan.arbaz"Zanotti, Pietro"https://zbmath.org/authors/?q=ai:zanotti.pietroThe paper analyzes the numerical method for Biot's model describing the elastic wave propagation inside a porous medium saturated with a fluid. Variables in this model represent the displacement of the medium and the fluid pressure. In addition, there are several material parameters. However, spurious oscillations or volumetric locking may occur for specific values of these parameters. The authors focus on overcoming this problem and propose a method that is robust in the sense that it is uniformly stable with respect to all parameters.
First, the authors establish a novel nonsymmetric variational setting, where the norm measuring the data is not dual to the norm for measuring the solution. Then, they show the well-posedness of the setting and derive stability estimates. Furthermore, the authors propose a method that uses the backward Euler scheme for temporal discretization combined with the finite element method using first-order nonconforming Crouzeix-Raviart elements for the displacement and first-order discontinuous piecewise affine functions for the fluid pressure. The presented analysis of stability and error estimates leads to the conclusion that the method is robust and quasi-optimal. Finally, possible generalizations of the results are discussed.
Reviewer: Dana Černá (Liberec)Second-order convergence of the linearly extrapolated Crank-Nicolson method for the Navier-Stokes equations with \(H^1\) initial datahttps://zbmath.org/1491.650972022-09-13T20:28:31.338867Z"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyang"Ma, Shu"https://zbmath.org/authors/?q=ai:ma.shu"Wang, Na"https://zbmath.org/authors/?q=ai:wang.naSummary: This article concerns the numerical approximation of the two-dimensional nonstationary Navier-Stokes equations with \(H^1\) initial data. By utilizing special locally refined temporal stepsizes, we prove that the linearly extrapolated Crank-Nicolson scheme, with the usual stabilized Taylor-Hood finite element method in space, can achieve second-order convergence in time and space. Numerical examples are provided to support the theoretical analysis.Local and parallel efficient BDF2 and BDF3 rotational pressure-correction schemes for a coupled Stokes/Darcy systemhttps://zbmath.org/1491.650982022-09-13T20:28:31.338867Z"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian.1"Wang, Xue"https://zbmath.org/authors/?q=ai:wang.xue"Al Mahbub, Md. Abdullah"https://zbmath.org/authors/?q=ai:al-mahbub.md-abdullah"Zheng, Haibiao"https://zbmath.org/authors/?q=ai:zheng.haibiao"Chen, Zhangxin"https://zbmath.org/authors/?q=ai:chen.zhangxinThis paper extends authors earlier work [\textit{J. Li} et al., Comput. Math. with Appl. 79, 337--353 (2020; Zbl 1443.65187); Numer. Methods Partial Differential Equations 35, 1873--1889 (2019; Zbl 1423.76253)] where first- and second-order (in time) BE (backward Euler) and BDF2 schemes with the rotational pressure-correction methods introduced in [\textit{J. Guermond} et al., SIAM J. Numer. Anal. 43, 239--258 (2005; Zbl 1083.76044)] are studied for a coupled Stokes/Darcy system. These temporal schemes are developed, and the BDF2/BDF3 rotational pressure-correction methods are studied for the Stokes/Darcy system. It was proven that the BDF2/BDF3 rotational pressure-correction methods are unconditionally stable, long-time accurate with a uniform-in-time error bound, and efficient in that only two decoupled equations are required to solve at each time step. At each time step, only one linear system of equations has to be solved, which thus significantly reduces the computational time and memory costs in practice. The presented projection methods are combined with the local and parallel methods based on full overlapping decoupled techniques for the coupled Stokes/Darcy system which increases the computational efficiency further. Several numerical examples are presented to illustrate the accuracy and efficiency of the proposed methods.
Reviewer: Bülent Karasözen (Ankara)A discontinuous Galerkin pressure correction scheme for the incompressible Navier-Stokes equations: stability and convergencehttps://zbmath.org/1491.650992022-09-13T20:28:31.338867Z"Masri, Rami"https://zbmath.org/authors/?q=ai:masri.rami"Liu, Chen"https://zbmath.org/authors/?q=ai:liu.chen"Riviere, Beatrice"https://zbmath.org/authors/?q=ai:riviere.beatrice-mSummary: A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier-Stokes equations is formulated and analyzed. We prove unconditional stability of the proposed scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates.Superconvergence error estimates of discontinuous Galerkin time stepping for singularly perturbed parabolic problemshttps://zbmath.org/1491.651002022-09-13T20:28:31.338867Z"Singh, Gautam"https://zbmath.org/authors/?q=ai:singh.gautam-b"Natesan, Srinivasan"https://zbmath.org/authors/?q=ai:natesan.srinivasanSummary: A parabolic convection-diffusion-reaction problem is discretized by the non-symmetric interior penalty Galerkin (NIPG) method in space and discontinuous Galerkin (DG) method in time. To improve the order of convergence of the numerical scheme, we have used piecewise Lagrange interpolation at Gauss points and estimated the error bound in the discrete energy norm. We have shown superconvergence properties of the DG method, i.e., \((k + 1)\)-order convergence in space and \((l + 1)\)-order convergence in time, where \(k\) and \(l\) are the degrees of piecewise polynomials in the finite element space used in spatial and temporal variables, respectively. Numerical results are given to verify our theoretical findings.New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous mediahttps://zbmath.org/1491.651012022-09-13T20:28:31.338867Z"Sun, Weiwei"https://zbmath.org/authors/?q=ai:sun.weiweiSummary: Numerical methods and analysis for compressible miscible flow in porous media have been investigated extensively in the last several decades. Amongst those methods, the lowest-order mixed method is the most popular one in practical applications. The method is based on the linear Lagrange approximation for the concentration and the lowest order (zero-order) Raviart-Thomas mixed approximation for the Darcy velocity/pressure. However, the existing error analysis only provides the first-order accuracy in \(L^2\)-norm for all three physical components in spatial direction, which was proved under certain extra restrictions on both time step and spatial meshes. The analysis is not optimal for the concentration mainly due to the strong coupling of the system and the drawback of the traditional approach which leads to serious pollution to the numerical concentration in analysis. The main task of this paper is to present a new analysis and establish the optimal error estimate of the commonly-used linearized lowest-order mixed FEM. In particular, the second-order accuracy for the concentration in spatial direction is proved unconditionally. Moreover, we propose a simple recovery technique to obtain a new numerical Darcy velocity/pressure of second-order accuracy by re-solving an elliptic pressure equation. Also we extend our analysis to a second-order time discrete scheme to obtain optimal error estimates in both spatial and temporal directions. Numerical results are provided to confirm our theoretical analysis and show the efficiency of the method.A fast distributed data-assimilation algorithm for divergence-free advectionhttps://zbmath.org/1491.651022022-09-13T20:28:31.338867Z"Tchrakian, Tigran T."https://zbmath.org/authors/?q=ai:tchrakian.tigran-t"Zhuk, Sergiy"https://zbmath.org/authors/?q=ai:zhuk.sergiy-mA fast data assimilation algorithm for a 2D linear advection equation with divergence-free coefficients is introduced. The authors apply the nodal discontinuous Galerkin (DG) method to discretize the advection equation. Then, they briefly discuss the distributed filtering and its intrinsic relation to the DG discretization. In the future, a convergence analysis should be given.
Reviewer: Zhen Chao (Milwaukee)Local error estimates for Runge-Kutta discontinuous Galerkin methods with upwind-biased numerical fluxes for a linear hyperbolic equation in one-dimension with discontinuous initial datahttps://zbmath.org/1491.651032022-09-13T20:28:31.338867Z"Xu, Yuan"https://zbmath.org/authors/?q=ai:xu.yuan.1|xu.yuan"Zhao, Di"https://zbmath.org/authors/?q=ai:zhao.di"Zhang, Qiang"https://zbmath.org/authors/?q=ai:zhang.qiang.2Summary: In this paper we present the local error estimate of Runge-Kutta discontinuous Galerkin (RKDG) methods with upwind-biased numerical fluxes for solving linear hyperbolic equations in one dimension with discontinuous initial data. Under the temporal-spatial condition to ensure the \(\mathrm{L}^2\)-norm stability, the convergence orders in both space and time are optimal outside the pollution region due to the discontinuous initial data, whose width is nearly optimal to be one-half order of the mesh size, no matter in the upwind or the downwind direction. The kernel analysis is achieving the weighted \(\mathrm{L}^2\)-norm stability result for the RKDG methods, based on the matrix transferring process and the local \(\mathrm{L}^2\)-projection. Furthermore, we need to set up the weighted approximation property for the generalized Gauss-Radau projection. Finally some numerical experiments are also given to support the theoretical results.A two-grid combined mixed finite element and discontinuous Galerkin method for an incompressible miscible displacement problem in porous mediahttps://zbmath.org/1491.651042022-09-13T20:28:31.338867Z"Yang, Jiming"https://zbmath.org/authors/?q=ai:yang.jiming"Su, Yifan"https://zbmath.org/authors/?q=ai:su.yifanSummary: An incompressible miscible displacement problem is investigated. A two-grid algorithm of a full-discretized combined mixed finite element and discontinuous Galerkin approximation to the miscible displacement in porous media is proposed. The error estimate for the concentration in \(H^1\)-norm and the error estimates for the pressure and the velocity in \(L^2\)-norm are obtained. The analysis shows that the asymptotically optimal approximation can be achieved as long as the mesh size satisfies \(h = O(H^2)\), where \(H\) and \(h\) are the sizes of the coarse mesh and the fine mesh, respectively. Meanwhile, the effectiveness of the presented algorithm is verified by numerical experiments, from which it can be seen that the algorithm is spent much less time.Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cellhttps://zbmath.org/1491.651052022-09-13T20:28:31.338867Z"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the ``zero-energy-contribution'' feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.Structure-preserving nonlinear filtering for continuous and discontinuous Galerkin spectral/\(hp\) element methodshttps://zbmath.org/1491.651062022-09-13T20:28:31.338867Z"Zala, Vidhi"https://zbmath.org/authors/?q=ai:zala.vidhi"Kirby, Robert M."https://zbmath.org/authors/?q=ai:kirby.robert-m-ii"Narayan, Akil"https://zbmath.org/authors/?q=ai:narayan.akil-cThe convergence analysis of semi- and fully-discrete projection-decoupling schemes for the generalized Newtonian modelshttps://zbmath.org/1491.651072022-09-13T20:28:31.338867Z"Zhou, Guanyu"https://zbmath.org/authors/?q=ai:zhou.guanyuSummary: We propose two linear schemes (1st- and 2nd-order) for the generalized Newtonian flow with the shear-dependent viscosity, which combine the decoupling techniques with the projection methods. The linear stabilization terms mimic \(-k\partial_t \Delta{\boldsymbol{u}}\) and \(-k\partial_{tt} \Delta{\boldsymbol{u}}\) from the PDE point of view. By our schemes, each velocity component can be computed in parallel efficiently using the same solver \((I-\alpha^{-1}k\Delta)^{-1}\) at every time level. We analyze the convergence rates of the (temporally) semi- and the fully-discrete schemes. The theoretical results are testified by the numerical experiments.Quadratic spline function for the approximate solution of an intermediate space-fractional advection diffusion equationhttps://zbmath.org/1491.651082022-09-13T20:28:31.338867Z"Abdel-Rehi, E. A."https://zbmath.org/authors/?q=ai:abdel-rehi.e-a"Brikaa, M. G."https://zbmath.org/authors/?q=ai:brikaa.m-gSummary: The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.On the numerical solution of some differential equations with nonlocal integral boundary conditions via Haar wavelethttps://zbmath.org/1491.651092022-09-13T20:28:31.338867Z"Aziz, Imran"https://zbmath.org/authors/?q=ai:aziz.imran"Nisar, Muhammad"https://zbmath.org/authors/?q=ai:nisar.muhammad-danish"Siraj-ul-Islam"https://zbmath.org/authors/?q=ai:siraj-ul-islam.Summary: Differential equations with nonlocal boundary conditions are used to model a number of physical phenomena encountered in situations where data on the boundary cannot be measured directly. This study explores numerical solutions to elliptic, parabolic and hyperbolic equations with two different types of nonlocal integral boundary conditions. The numerical solutions are obtained using the Haar wavelet collocation method with the aid of Finite Differences for time derivatives. The method is applicable to both linear and nonlinear problems. To obtain the numerical solutions, Gauss elimination method is used for linear and Newton's method for nonlinear differential equations. The validity of the proposed method is demonstrated by solving several benchmark test problems from the literature: two elliptic linear and two nonlinear samples covering both types of nonlocal integral boundary conditions; one nonlinear and two linear test problems for parabolic partial differential equations; two linear samples for hyperbolic partial differential equations. The accuracy of the method is verified by comparing the numerical results with the analytical solutions. The numerical results confirm that the method is simple and effective.A new Lagrange multiplier approach for constructing structure preserving schemes. II: Bound preservinghttps://zbmath.org/1491.651102022-09-13T20:28:31.338867Z"Cheng, Qing"https://zbmath.org/authors/?q=ai:cheng.qing"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jieIn this paper, positivity preserving schemes using Lagrange multiplier approach in Part I [\textit{Q. Cheng} and \textit{J. Shen}, Comput. Methods Appl. Mech. Eng. 391, Article ID 114585, 25 p. (2022; Zbl 07487691)] are extended to construct bound preserving schemes for a class of nonlinear PDEs in the following form:
\[
u_t + {\mathcal L }u + {\mathcal N }(u) = 0
\]
with suitable initial and boundary conditions, where \({\mathcal L}\) is a linear or nonlinear nonnegative operator and \({\mathcal N}(u)\) is a semilinear or quasi-linear operator.
For problems which also conserve mass, these bound preserving schemes are modified to also conserve mass. For second-order parabolic-type equations stability results are established with a second-order scheme with mass conservation. A hybrid spectral method is considered for error analysis for a fully discretized second-order scheme. These schemes are applied to several typical PDEs with bound and/or mass preserving properties and are validated for a variety of problems with bound preserving solutions, including the Allen-Cahn and Cahn-Hilliard equations and a class of Fokker-Planck equations. The schemes constructed in this paper include the cutoff approach [\textit{C. Lu} et al., J. Computat. Phys., 242, 24--36 (2013; Zbl 1297.65097)] as a special case, so that they provide an alternative interpretation of the cutoff approach and allows us to construct new cutoff implicit-explicit (IMEX) schemes with mass conservation.
Reviewer: Bülent Karasözen (Ankara)Second-order SAV schemes for the nonlinear Schrödinger equation and their error analysishttps://zbmath.org/1491.651112022-09-13T20:28:31.338867Z"Deng, Beichuan"https://zbmath.org/authors/?q=ai:deng.beichuan"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jie"Zhuang, Qingqu"https://zbmath.org/authors/?q=ai:zhuang.qingquSummary: We consider a second-order SAV scheme for the nonlinear Schrödinger equation in the whole space with typical generalized nonlinearities, and carry out a rigorous error analysis. We also develop a fully discretized SAV scheme with Hermite-Galerkin approximation for the space variables, and present numerical experiments to validate our theoretical results.A novel discrete fractional Grönwall-type inequality and its application in pointwise-in-time error estimateshttps://zbmath.org/1491.651122022-09-13T20:28:31.338867Z"Li, Dongfang"https://zbmath.org/authors/?q=ai:li.dongfang"She, Mianfu"https://zbmath.org/authors/?q=ai:she.mianfu"Sun, Hai-wei"https://zbmath.org/authors/?q=ai:sun.haiwei"Yan, Xiaoqiang"https://zbmath.org/authors/?q=ai:yan.xiaoqiangSummary: We present a family of fully-discrete schemes for numerically solving nonlinear sub-diffusion equations, taking the weak regularity of the exact solutions into account. Using a novel discrete fractional Grönwall inequality, we obtain pointwise-in-time error estimates of the time-stepping methods. It is proved that as \(t\rightarrow 0\), the convergence orders can be \(\sigma_k\), where \(\sigma_k\) is the regularity parameter. The initial convergence results are sharp. As \(t\) is far away from 0, the schemes give a better convergence results. Numerical experiments are given to confirm the theoretical results.The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applicationshttps://zbmath.org/1491.651132022-09-13T20:28:31.338867Z"Pourbabaee, Marzieh"https://zbmath.org/authors/?q=ai:pourbabaee.marzieh"Saadatmandi, Abbas"https://zbmath.org/authors/?q=ai:saadatmandi.abbasSummary: In this paper, the properties of Chebyshev polynomials and the Gauss-Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given.Analysis of the Fitzhugh Nagumo model with a new numerical schemehttps://zbmath.org/1491.651142022-09-13T20:28:31.338867Z"Mishra, Jyoti"https://zbmath.org/authors/?q=ai:mishra.jyotiSummary: The model describing a prototype of an excitable system was extended using the newly established concept of fractional differential operators with non-local and non-singular kernel in this paper. We presented a detailed discussion underpinning the well-poseness of the extended model. Due to the non-linearity of the modified model, we solved it using a newly established numerical scheme for partial differential equations that combines the fundamental theorem of fractional calculus, the Laplace transform and the Lagrange interpolation approximation. We presented some numerical simulations that, of course reflect asymptotically the real world observed behaviors.A highly accurate difference method for solving the Dirichlet problem for Laplace's equation on a rectanglehttps://zbmath.org/1491.651152022-09-13T20:28:31.338867Z"Dosiyev, Adiguzel A."https://zbmath.org/authors/?q=ai:dosiyev.adiguzel-a"Sarikaya, Hediye"https://zbmath.org/authors/?q=ai:sarikaya.hediyeSummary: \(O(h^8)\) order \((h\) is the mesh size) of accurate three-stage difference method on a square grid for the approximate solution of the Dirichlet problem for Laplace's equation on a rectangle is proposed and justified without taking more than 9 nodes of the grid. At the first stage, by using the 9-point scheme the sum of the pure fourth derivatives of the desired solution is approximated of order \(O(h^6)\). At the second stage, approximate values of the sum of the pure eighth derivatives is approximated of order \(O(h^2)\) by the 5-point scheme. At the final third stage, the system of simplest 5-point difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first and second stages. Numerical experiment is illustrated to support the analysis made.
For the entire collection see [Zbl 1436.46003].A numerical algorithm to computationally solve the Hemker problem using Shishkin mesheshttps://zbmath.org/1491.651162022-09-13T20:28:31.338867Z"Hegarty, A. F."https://zbmath.org/authors/?q=ai:hegarty.alan-f"O'Riordan, E."https://zbmath.org/authors/?q=ai:oriordan.eugeneA numerical algorithm is presented to solve a benchmark problem proposed by \textit{P. W. Hemker} [J. Comput. Appl. Math. 76, No. 1--2, 277--285 (1996; Zbl 0870.35020)]. The numerical algorithm is composed of several different Shishkin meshes defined across different coordinate systems aligned to three overlapping subdomains. It is constructed based on parameter explicit pointwise bounds on how the continuous solutions decay away from the circle. Using upwinding in all co-ordinate directions, numerical solutions exhibit no spurious oscillations. Several layer-adapted Shishkin meshes are utilized and these grids are aligned both to the geometry of the domain and to the dominant direction of decay within the boundary/interior layer functions. Numerical experiments indicate that the method is producing accurate approximations over an extensive range of the singular perturbation parameter. The numerical approximations are converging to the continuous solution for each value of the parameter, but not uniformly in the singular perturbation parameter
Reviewer: Bülent Karasözen (Ankara)Convergence rate analysis for deep Ritz methodhttps://zbmath.org/1491.651172022-09-13T20:28:31.338867Z"Duan, Chenguang"https://zbmath.org/authors/?q=ai:duan.chenguang"Jiao, Yuling"https://zbmath.org/authors/?q=ai:jiao.yuling"Lai, Yanming"https://zbmath.org/authors/?q=ai:lai.yanming"Li, Dingwei"https://zbmath.org/authors/?q=ai:li.dingwei"Lu, Xiliang"https://zbmath.org/authors/?q=ai:lu.xiliang"Yang, Jerry Zhijian"https://zbmath.org/authors/?q=ai:yang.jerry-zhijianSummary: Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) [\textit{W. E} and \textit{B. Yu}, Commun. Math. Stat. 6, No. 1, 1--12 (2018; Zbl 1392.35306)] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in \(H^1\) norm for DRM using deep networks with \(\mathrm{ReLU}^2\) activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bound on the approximation error of deep \(\mathrm{ReLU}^2\) network in \(C^1\) norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and \(\mathrm{ReLU}^2\) network, both of which are of independent interest.A rate of convergence of physics informed neural networks for the linear second order elliptic PDEshttps://zbmath.org/1491.651182022-09-13T20:28:31.338867Z"Jiao, Yuling"https://zbmath.org/authors/?q=ai:jiao.yuling"Lai, Yanming"https://zbmath.org/authors/?q=ai:lai.yanming"Li, Dingwei"https://zbmath.org/authors/?q=ai:li.dingwei"Lu, Xiliang"https://zbmath.org/authors/?q=ai:lu.xiliang"Wang, Fengru"https://zbmath.org/authors/?q=ai:wang.fengru"Wang, Yang"https://zbmath.org/authors/?q=ai:wang.yang.1|wang.yang"Yang, Jerry Zhijian"https://zbmath.org/authors/?q=ai:yang.jerry-zhijianSummary: In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in \(C^2\) norm with \(\mathrm{ReLU}^3\) networks (deep network with activation function \(\max\{0, x^3\})\) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with \(\mathrm{ReLU}^3\) network, which is of immense independent interest.An adaptive finite element DtN method for the elastic wave scattering problemhttps://zbmath.org/1491.651192022-09-13T20:28:31.338867Z"Li, Peijun"https://zbmath.org/authors/?q=ai:li.peijun.1"Yuan, Xiaokai"https://zbmath.org/authors/?q=ai:yuan.xiaokaiSummary: Consider the scattering of an incident wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.Ensemble Kalman filter for neural network-based one-shot inversionhttps://zbmath.org/1491.651202022-09-13T20:28:31.338867Z"Guth, Philipp A."https://zbmath.org/authors/?q=ai:guth.philipp-a"Schillings, Claudia"https://zbmath.org/authors/?q=ai:schillings.claudia"Weissmann, Simon"https://zbmath.org/authors/?q=ai:weissmann.simonSummary: We study the use of novel techniques arising in machine learning for inverse problems. Our approach replaces the complex forward model by a neural network, which is trained simultaneously in a one-shot sense when estimating the unknown parameters from data, i.e., the neural network is trained only for the unknown parameter. By establishing a link to the Bayesian approach to inverse problems we develop an algorithmic framework that ensures the feasibility of the parameter estimate with respect to the forward model. We propose an efficient, derivative-free optimization method based on variants of the ensemble Kalman inversion. Numerical experiments show that the ensemble Kalman filter for neural network-based one-shot inversion is a promising direction combining optimization and machine learning techniques for inverse problems.
For the entire collection see [Zbl 1491.49002].A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaceshttps://zbmath.org/1491.651212022-09-13T20:28:31.338867Z"Hu, Guanghui"https://zbmath.org/authors/?q=ai:hu.guanghui"Liu, Xiaoli"https://zbmath.org/authors/?q=ai:liu.xiaoli"Zhang, Bo"https://zbmath.org/authors/?q=ai:zhang.bo"Zhang, Haiwen"https://zbmath.org/authors/?q=ai:zhang.haiwenUnique continuation on quadratic curves for harmonic functionshttps://zbmath.org/1491.651222022-09-13T20:28:31.338867Z"Ke, Yufei"https://zbmath.org/authors/?q=ai:ke.yufei"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.7|chen.yu.2|chen.yu.5|chen.yu.8|chen.yuqun|chen.yu.4|chen.yu.1|chen.yu.3|chen.yu.6Summary: The unique continuation on quadratic curves for harmonic functions is discussed in this paper. By using complex extension method, the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated. The numerical algorithm is provided based on collocation method and Tikhonov regularization. The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.An efficient DWR-type a posteriori error bound of SDFEM for singularly perturbed convection-diffusion PDEshttps://zbmath.org/1491.651232022-09-13T20:28:31.338867Z"Avijit, D."https://zbmath.org/authors/?q=ai:avijit.d"Natesan, S."https://zbmath.org/authors/?q=ai:natesan.srinivasanSummary: This article deals with the residual-based a posteriori error estimation in the standard energy norm for the streamline-diffusion finite element method (SDFEM) for singularly perturbed convection-diffusion equations. The well-known dual-weighted residual (DWR) technique has been adopted to elevate the accuracy of the error estimator. Our main contribution is finding an efficient computable DWR-type robust residual-based a posteriori error bound for the SDFEM. The local lower error bound has also been provided. An adaptive mesh refinement algorithm has been addressed and lastly, some numerical experiments are carried out to justify the theoretical proofs.Error estimates for a class of discontinuous Galerkin methods for nonsmooth problems via convex duality relationshttps://zbmath.org/1491.651242022-09-13T20:28:31.338867Z"Bartels, Sören"https://zbmath.org/authors/?q=ai:bartels.sorenSummary: We devise and analyze a class of interior penalty discontinuous Galerkin methods for nonlinear and nonsmooth variational problems. Discrete duality relations are derived that lead to optimal error estimates in the case of total-variation regularized minimization or obstacle problems. The analysis provides explicit estimates that precisely determine the role of stabilization parameters. Numerical experiments confirm the optimality of the estimates.A polygonal discontinuous Galerkin method with minus one stabilizationhttps://zbmath.org/1491.651252022-09-13T20:28:31.338867Z"Bertoluzza, Silvia"https://zbmath.org/authors/?q=ai:bertoluzza.silvia"Prada, Daniele"https://zbmath.org/authors/?q=ai:prada.danieleSummary: We propose a discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each element \(K\), a residual term involving the fluxes, measured in the norm of the dual of \(H^1 (K)\). The scalar product corresponding to such a norm is numerically realized \textit{via} the introduction of a (minimal) auxiliary space inspired by the Virtual Element Method. Stability and optimal error estimates in the broken \(H^1 \) norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flowhttps://zbmath.org/1491.651262022-09-13T20:28:31.338867Z"Burman, Erik"https://zbmath.org/authors/?q=ai:burman.erik"Puppi, Riccardo"https://zbmath.org/authors/?q=ai:puppi.riccardoSummary: We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as \(\mathcal{O} (h^{-1})\), while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as \(\mathcal{O} (h^{- k -1})\), \(k\) being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual \(L^2\)-norm. However, we are still able to recover the optimal a priori \(L^2\)-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.Convergence of anisotropic mesh adaptation via metric optimizationhttps://zbmath.org/1491.651272022-09-13T20:28:31.338867Z"Carson, Hugh A."https://zbmath.org/authors/?q=ai:carson.hugh-a"Allmaras, Steven"https://zbmath.org/authors/?q=ai:allmaras.steven-r"Galbraith, Marshall"https://zbmath.org/authors/?q=ai:galbraith.marshall-c"Darmofal, David"https://zbmath.org/authors/?q=ai:darmofal.david-lThis article discusses the convergence for a class of metric adaptive finite element method, utilizing an optimization statement to relate the error on the sequence of meshes. In addition, the authors prove that such a sequence of meshes lead to the optimal asymptotic rate of convergence for a given polynomial order. Numerical results ilcluded in the article support the performance of the algorithm for a singularly perturbed linear advection diffusion problem.
Reviewer: Marius Ghergu (Dublin)Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEshttps://zbmath.org/1491.651282022-09-13T20:28:31.338867Z"Carstensen, C."https://zbmath.org/authors/?q=ai:carstensen.carsten"Nataraj, Neela"https://zbmath.org/authors/?q=ai:nataraj.neela"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumarSummary: For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients \({\mathbf{A}}, {\mathbf{b}},\gamma\) in \(L^\infty\) and symmetric and uniformly positive definite coefficient matrix \({\mathbf{A}} \), this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in \(H(\mathrm{div})\times L^2\) as well as in in \(L^2\times L^2\) up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to \(L^\infty\) coefficients. The compactness argument of \textit{A. H. Schatz} and \textit{J. Wang} [Math. Comput. 65, No. 213, 19--27 (1996; Zbl 0856.65129)] for the displacement-oriented problem does \textit{not} apply immediately to the mixed formulation in \(H(\mathrm{div})\times L^2\). But it allows the uniform approximation of some \(L^2\) contributions and can be combined with a recent \(L^2\) best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equationshttps://zbmath.org/1491.651292022-09-13T20:28:31.338867Z"Caucao, Sergio"https://zbmath.org/authors/?q=ai:caucao.sergio"Gatica, Gabriel N."https://zbmath.org/authors/?q=ai:gatica.gabriel-n"Ortega, Juan P."https://zbmath.org/authors/?q=ai:ortega.juan-pabloSummary: We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman-Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška-Brezzi's theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimal \textit{a priori} error estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.Analysis of a stabilized finite element approximation for a linearized logarithmic reformulation of the viscoelastic flow problemhttps://zbmath.org/1491.651302022-09-13T20:28:31.338867Z"Codina, Ramon"https://zbmath.org/authors/?q=ai:codina.ramon"Moreno, Laura"https://zbmath.org/authors/?q=ai:moreno.lauraSummary: In this paper we present the numerical analysis of a finite element method for a linearized viscoelastic flow problem. In particular, we analyze a linearization of the logarithmic reformulation of the problem, which in particular should be able to produce results for Weissenberg numbers higher than the standard one. In order to be able to use the same interpolation for all the unknowns (velocity, pressure and logarithm of the conformation tensor), we employ a stabilized finite element formulation based on the Variational Multi-Scale concept. The study of the linearized problem already serves to show why the logarithmic reformulation performs better than the standard one for high Weissenberg numbers; this is reflected in the stability and error estimates that we provide in this paper.A reduced basis method for fractional diffusion operators. IIhttps://zbmath.org/1491.651312022-09-13T20:28:31.338867Z"Danczul, Tobias"https://zbmath.org/authors/?q=ai:danczul.tobias"Schöberl, Joachim"https://zbmath.org/authors/?q=ai:schoberl.joachimSummary: We present a novel numerical scheme to approximate the solution map \(s \mapsto u(s) := \mathcal{L}^{-s} f\) to fractional PDEs involving elliptic operators. Reinterpreting \(\mathcal{L}^{-s}\) as an interpolation operator allows us to write \(u(s)\) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation \(L\) of the operator whose inverse is projected to the \(s\)-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously.
A second algorithm is presented to avoid inversion of \(L\). Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.
For Part I, see [\textit{T. Danczul} and \textit{J. Schöberl}, Numer. Math. 151, No. 2, 369--404 (2022; Zbl 07536676)].New \(H(\mathrm{div})\)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal mesheshttps://zbmath.org/1491.651322022-09-13T20:28:31.338867Z"Devloo, Philippe R. B."https://zbmath.org/authors/?q=ai:devloo.philippe-remy-bernard"Farias, Agnaldo M."https://zbmath.org/authors/?q=ai:farias.agnaldo-m"Gomes, Sônia M."https://zbmath.org/authors/?q=ai:gomes.sonia-maria"Pereira, Weslley"https://zbmath.org/authors/?q=ai:pereira.weslley-s"Dos Santos, Antonio J. B."https://zbmath.org/authors/?q=ai:dos-santos.antonio-j-b"Valentin, Frédéric"https://zbmath.org/authors/?q=ai:valentin.fredericSummary: This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforced \textit{via} the rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in the \(H(\mathrm{div})\) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in the \(L^2\)-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.Non-symmetric isogeometric FEM-BEM couplingshttps://zbmath.org/1491.651332022-09-13T20:28:31.338867Z"Elasmi, Mehdi"https://zbmath.org/authors/?q=ai:elasmi.mehdi"Erath, Christoph"https://zbmath.org/authors/?q=ai:erath.christoph"Kurz, Stefan"https://zbmath.org/authors/?q=ai:kurz.stefanSummary: We present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting of two disjoint domains. We consider the Finite Element Method in the bounded domains to simulate possibly non-linear materials. The Boundary Element Method is applied in unbounded or thin domains where the material behavior is linear. The isogeometric framework allows to combine different design and analysis tools: first, we consider the same type of NURBS parameterizations for an exact geometry representation and second, we use the numerical analysis for the Galerkin approximation. Moreover, it facilitates to perform \(h\)- and \(p\)-refinements. For the sake of analysis, we consider the framework of strongly monotone and Lipschitz continuous operators to ensure well-posedness of the coupled system. Furthermore, we provide a priori error estimates. We additionally show an improved convergence behavior for the errors in functionals of the solution that may double the rate under certain assumptions. Numerical examples conclude the work which illustrate the theoretical results.An efficient iterative method for solving parameter-dependent and random convection-diffusion problemshttps://zbmath.org/1491.651342022-09-13T20:28:31.338867Z"Feng, Xiaobing"https://zbmath.org/authors/?q=ai:feng.xiaobing"Luo, Yan"https://zbmath.org/authors/?q=ai:luo.yan"Vo, Liet"https://zbmath.org/authors/?q=ai:vo.liet"Wang, Zhu"https://zbmath.org/authors/?q=ai:wang.zhuSummary: This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection-diffusion problems. It is inspired by the multi-modes method and the ensemble method and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent convection and diffusion coefficients and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the \textit{LU} matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion and convection-diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.Singularly perturbed reaction-diffusion problems as first order systemshttps://zbmath.org/1491.651352022-09-13T20:28:31.338867Z"Franz, Sebastian"https://zbmath.org/authors/?q=ai:franz.sebastianSummary: We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and \(H_{\mathrm{div}}\)-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced \(H^2\)-norm for the second order formulation.MINRES for second-order PDEs with singular datahttps://zbmath.org/1491.651362022-09-13T20:28:31.338867Z"Führer, Thomas"https://zbmath.org/authors/?q=ai:fuhrer.thomas"Heuer, Norbert"https://zbmath.org/authors/?q=ai:heuer.norbert"Karkulik, Michael"https://zbmath.org/authors/?q=ai:karkulik.michaelIn this work, the minimum residual finite element methods (MINRES FEM) are analyzed with source functionals in \(H^{-1}(\Omega )\), the dual space of \(H^1_0 (\Omega )\) with \(\Omega \subset \mathbb{R}^d\) (\(d = 2, 3\)) a polytopal domain, and point sources by focusing on the first-order reformulation of the Poisson problem. The discontinuous Petrov-Galerkin (DPG) method with optimal test functions is also studied. They are MINRES methods that minimize a functional in the dual norm of broken test spaces. The results of the paper are also extended and analyzed for the DPG method for such data. The authors show that the MINRES FEM for the Poisson problem on Lipschitz domains can be modified to handle \(H^{-1}\) loads and lead to optimal convergence rates. The regularization approaches are extended that allow the use of \(H^{-1}\) loads and to the case of point loads. Appropriate convergence orders are proven for all cases. Several numerical experiments are presented that confirm the theoretical results. The approach can also be extended to general well-posed second-order problems.
Reviewer: Bülent Karasözen (Ankara)An embedded discontinuous Galerkin method for the Oseen equationshttps://zbmath.org/1491.651372022-09-13T20:28:31.338867Z"Han, Yongbin"https://zbmath.org/authors/?q=ai:han.yongbin"Hou, Yanren"https://zbmath.org/authors/?q=ai:hou.yanrenSummary: In this paper, the \textit{a priori} error estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented. It is proved that the velocity error in the \(L^2 (\Omega)\) norm, has an optimal error bound with convergence order \(k+1\), where the constants are dependent on the Reynolds number (or \(\nu^{-1})\), in the diffusion-dominated regime, and in the convection-dominated regime, it has a Reynolds-robust error bound with quasi-optimal convergence order \(k+1/2\). Here, \(k\) is the polynomial order of the velocity space. In addition, we also prove an optimal error estimate for the pressure. Finally, we carry out some numerical experiments to corroborate our analytical results.Numerical upscaling for heterogeneous materials in fractured domainshttps://zbmath.org/1491.651382022-09-13T20:28:31.338867Z"Hellman, Fredrik"https://zbmath.org/authors/?q=ai:hellman.fredrik"Målqvist, Axel"https://zbmath.org/authors/?q=ai:malqvist.axel"Wang, Siyang"https://zbmath.org/authors/?q=ai:wang.siyangSummary: We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise \textit{e.g.} in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An \textit{a priori} error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.Optimal maximum norm estimates for virtual element methodshttps://zbmath.org/1491.651392022-09-13T20:28:31.338867Z"He, Wen-Ming"https://zbmath.org/authors/?q=ai:he.wenming"Guo, Hailong"https://zbmath.org/authors/?q=ai:guo.hailongThis article discusses the maximum norm error estimations for virtual element methods. To this aim, the authors obtain higher local regularity results based on the analysis of Green's functions and high-order local error estimations for the partition of the virtual element solutions. The maximum norm of the exact gradient and the gradient of the projection of the virtual element solutions are proved to achieve optimal convergence results. For high-order virtual element methods, the authors establish the optimal convergence results in \(L^\infty\) norm. Several numerical experiments on general polygonal meshes are included to support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problemshttps://zbmath.org/1491.651402022-09-13T20:28:31.338867Z"He, Xiaoxiao"https://zbmath.org/authors/?q=ai:he.xiaoxiao"Song, Fei"https://zbmath.org/authors/?q=ai:song.fei"Deng, Weibing"https://zbmath.org/authors/?q=ai:deng.weibingThe paper deals with the numerical solution of the Stokes interface problem by a stabilized extended finite element method on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming \(P_1\) velocity and elementwise \(P_0\) pressure. Extra stabilization terms involving velocity and pressure are added in the discrete bilinear form. An inf-sup stability result is derived, which is uniform with respect to mesh size \(h\), the viscosity and the position of the interface. An optimal priori error estimates are obtained. Moreover, the errors in energy norm for velocity and in \(L_2\) norm for pressure are uniform to the viscosity and the location of the interface. Two numerical examples are presented to support the theoretical analysis.
Reviewer: Vit Dolejsi (Praha)A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral gridshttps://zbmath.org/1491.651412022-09-13T20:28:31.338867Z"Hu, Jun"https://zbmath.org/authors/?q=ai:hu.jun.4"Ma, Rui"https://zbmath.org/authors/?q=ai:ma.rui"Zhang, Min"https://zbmath.org/authors/?q=ai:zhang.min|zhang.min.5|zhang.min.6|zhang.min.2|zhang.min.1|zhang.min.3|zhang.min.7|zhang.min.4Summary: This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress \(\boldsymbol{\sigma} = - \nabla^2 u\) is sought in the Sobolev space \(H(\mathrm{div}\mathbf{div}, \Omega; \mathbb{S})\) simultaneously with the displacement \(u\) in \(L^2 (\Omega)\). By stemming from the structure of \(H(\mathbf{div}, \Omega; \mathbb{S})\) conforming elements for the linear elasticity problems proposed by \textit{J. Hu} and \textit{S. Zhang} [``A family of conforming mixed finite elements for linear elasticity on triangular grids'', Preprint, \url{arXiv:1406.7457}], the \(H(\mathrm{div}\mathbf{div}, \Omega; \mathbb{S})\) conforming finite element spaces are constructed by imposing the normal continuity of \(\mathbf{div}\boldsymbol{\sigma}\) on the \(H (\mathbf{div}, \Omega; \mathbb{S})\) conforming spaces of \(P_k\) symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for \(u\) are composed of the piecewise \(P_{k-2}\) polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for \(k \geqslant 3\), and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis.Error analysis of higher order trace finite element methods for the surface Stokes equationhttps://zbmath.org/1491.651422022-09-13T20:28:31.338867Z"Jankuhn, Thomas"https://zbmath.org/authors/?q=ai:jankuhn.thomas"Olshanskii, Maxim A."https://zbmath.org/authors/?q=ai:olshanskii.maxim-a"Reusken, Arnold"https://zbmath.org/authors/?q=ai:reusken.arnold"Zhiliakov, Alexander"https://zbmath.org/authors/?q=ai:zhiliakov.alexanderSummary: The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in \(\mathbb{R}^3\). The method employs parametric \(\mathbf{P}_k\)-\(P_{k-1}\) finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin-Helmholtz instability problem on the unit sphere.An immersed hybrid difference method for the elliptic interface equationhttps://zbmath.org/1491.651432022-09-13T20:28:31.338867Z"Jeon, Youngmok"https://zbmath.org/authors/?q=ai:jeon.youngmokThe author proposes an immersed hybrid difference method for elliptic interface problems. The method can be extended to three-dimensional problems easily. Numerical analysis should give more details in the future.
Reviewer: Zhen Chao (Milwaukee)Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined meshhttps://zbmath.org/1491.651442022-09-13T20:28:31.338867Z"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyangSummary: The Galerkin finite element solution \(u_h\) of the Poisson equation \(-\Delta u=f\) under the Neumann boundary condition in a possibly nonconvex polygon \(\varOmega\), with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability:
\[
\|u_h\|_{L^{\infty}(\varOmega)} \le C\ell_h\|u\|_{L^{\infty}(\varOmega)},
\]
where \(\ell_h = \ln (2+1/h)\) for piecewise linear elements and \(\ell_h=1\) for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds:
\[
\|u-u_h\|_{L^{\infty}(\varOmega)} \le C\ell_h\|u-I_hu\|_{L^{\infty}(\varOmega)},
\]
where \(I_h\) denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm:
\[
\|u-u_h\|_{L^\infty (\varOmega)} \le
\begin{cases}
C\ell_h h^{k+2-\frac{2}{p}} \|f\|_{W^{k,p}(\varOmega)} &\text{if } r\ge k+1,\\
C\ell_h h^{k+1} \|f\|_{H^k(\varOmega)} &\text{if } r=k,
\end{cases}
\]
where \(r\ge 1\) is the degree of finite elements, \(k\) is any nonnegative integer no larger than \(r\), and \(p\in [2,\infty)\) can be arbitrarily large.High order unconditionally energy stable RKDG schemes for the Swift-Hohenberg equationhttps://zbmath.org/1491.651452022-09-13T20:28:31.338867Z"Liu, Hailiang"https://zbmath.org/authors/?q=ai:liu.hailiang"Yin, Peimeng"https://zbmath.org/authors/?q=ai:yin.peimengSummary: We propose unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows including the Swift-Hohenberg equation. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved for arbitrary time steps and spatial meshes. The method integrates a penalty free DG method for spatial discretization with a multi-stage algebraically stable RK method for temporal discretization by the energy quadratiztion (EQ) strategy. The resulting fully discrete DG method is proven to be unconditionally energy stable. By numerical tests on several benchmark problems we demonstrate the high order accuracy, energy stability, and simplicity of the proposed algorithm.Quasi-optimal adaptive hybridized mixed finite element methods for linear elasticityhttps://zbmath.org/1491.651462022-09-13T20:28:31.338867Z"Li, Yuwen"https://zbmath.org/authors/?q=ai:li.yuwenSummary: For the planar Navier-Lamé equation in mixed form with symmetric stress tensors, we prove the uniform quasi-optimal convergence of an adaptive method based on the hybridized mixed finite element proposed in [\textit{S. Gong} et al., Numer. Math. 141, No. 2, 569--604 (2019; Zbl 1412.65211)]. The main ingredients in the analysis consist of a discrete \textit{a posteriori} upper bound and a quasi-orthogonality result for the stress field under the mixed boundary condition. Compared with existing adaptive methods, the proposed adaptive algorithm could be directly applied to the traction boundary condition and be easily implemented.Solving the biharmonic plate bending problem by the Ritz method using explicit formulas for splines of degree 5https://zbmath.org/1491.651472022-09-13T20:28:31.338867Z"Lytvyn, O. M."https://zbmath.org/authors/?q=ai:lytvyn.oleg-m"Lytvyn, O. O."https://zbmath.org/authors/?q=ai:lytvyn.oleg-o"Tomanova, I. S."https://zbmath.org/authors/?q=ai:tomanova.i-sSummary: An algorithm is presented to solve the biharmonic problem for clamped plate by the Ritz method, with the use of explicit formulas for splines of degree 5 on a triangular grid of nodes. A computational experiment is carried out to solve the problem on a square domain for various partition schemes.Multiscale scattering in nonlinear Kerr-type mediahttps://zbmath.org/1491.651482022-09-13T20:28:31.338867Z"Maier, Roland"https://zbmath.org/authors/?q=ai:maier.roland|maier.roland.1"Verfürth, Barbara"https://zbmath.org/authors/?q=ai:verfurth.barbaraSummary: We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.Projection in negative norms and the regularization of rough linear functionalshttps://zbmath.org/1491.651492022-09-13T20:28:31.338867Z"Millar, F."https://zbmath.org/authors/?q=ai:millar.felipe"Muga, I."https://zbmath.org/authors/?q=ai:muga.ignacio"Rojas, S."https://zbmath.org/authors/?q=ai:rojas.sergio"van der Zee, K. G."https://zbmath.org/authors/?q=ai:van-der-zee.kristoffer-georgeSummary: In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as \(W_0^{1,q}(\varOmega)\), where \(1<q<\infty\) and \(\varOmega\) is a Lipschitz domain, we propose a projection method in negative Sobolev spaces \(W^{-1,p}(\varOmega), p\) being the conjugate exponent satisfying \(p^{-1} + q^{-1} = 1\). Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of \(W^{-1,p}(\varOmega)\), though not of \(L^1(\varOmega)\), but one strives for a regular approximation in \(L^1(\varOmega)\). We focus on projections onto discrete finite element spaces \(G_n\), and consider both discontinuous as well as continuous piecewise-polynomial approximations. While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspace \(V_m\). We show that this idea leads to a fully discrete method given by a mixed problem on \(V_m\times G_n\). We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods. We present numerical experiments that compute finite element approximations to Dirac delta's and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data.Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scatteringhttps://zbmath.org/1491.651502022-09-13T20:28:31.338867Z"Sheng, Qiwei"https://zbmath.org/authors/?q=ai:sheng.qiwei"Hauck, Cory D."https://zbmath.org/authors/?q=ai:hauck.cory-dSummary: We present an error analysis for the discontinuous Galerkin (DG) method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation with isotropic scattering. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter \(\varepsilon\) which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.The role of mesh quality and mesh quality indicators in the virtual element methodhttps://zbmath.org/1491.651512022-09-13T20:28:31.338867Z"Sorgente, T."https://zbmath.org/authors/?q=ai:sorgente.tommaso"Biasotti, S."https://zbmath.org/authors/?q=ai:biasotti.silvia"Manzini, G."https://zbmath.org/authors/?q=ai:manzini.gianmarco"Spagnuolo, M."https://zbmath.org/authors/?q=ai:spagnuolo.michelaSummary: Since its introduction, the virtual element method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the finite element method (FEM) analysis. In this work, we experimentally show that the VEM still converges, with almost optimal rates and low errors in the \(L^2, H^1\) and \(L^{\infty}\) norms, even if we significantly break the regularity assumptions that are used in the literature. These results suggest that the regularity assumptions proposed so far might be overestimated. We also exhibit examples on which the VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality indicator that experimentally correlates the entity of the violation of the regularity assumptions and the performance of the VEM solution, thus predicting if a mesh is potentially critical for VEM.A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous mediahttps://zbmath.org/1491.651522022-09-13T20:28:31.338867Z"Viguerie, Alex"https://zbmath.org/authors/?q=ai:viguerie.alex"Bertoluzza, Silvia"https://zbmath.org/authors/?q=ai:bertoluzza.silvia"Veneziani, Alessandro"https://zbmath.org/authors/?q=ai:veneziani.alessandro"Auricchio, Ferdinando"https://zbmath.org/authors/?q=ai:auricchio.ferdinandoAuthors' abstract: Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.
Reviewer: Wei Gong (Beijing)Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2Dhttps://zbmath.org/1491.651532022-09-13T20:28:31.338867Z"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.3|zhang.jin|zhang.jin.1|zhang.jin.2"Lv, Yanhui"https://zbmath.org/authors/?q=ai:lv.yanhuiSummary: On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method for singularly perturbed elliptic problems with two parameters on the unit square. It is a very challenging task to analyze uniform convergence of finite element method on this mesh in 2D. The existing analysis tool, quasi-interpolation, is only applicable to one-dimensional case because of the complexity of Bakhvalov-type mesh in 2D. In this paper, a powerful tool, Lagrange-type interpolation, is proposed, which is simple and effective and can be used in both 1D and 2D. The application of this interpolation in 2D must be handled carefully. Some boundary correction terms must be introduced to maintain the homogeneous Dirichlet boundary condition. These correction terms are difficult to be handled because the traditional analysis do not work for them. To overcome this difficulty, we derive a delicate estimation of the width of some mesh. Moreover, we adopt different analysis strategies for different layers. Finally, we prove uniform convergence of optimal order. Numerical results verify the theoretical analysis.Coupled iterative analysis for stationary inductionless magnetohydrodynamic system based on charge-conservative finite element methodhttps://zbmath.org/1491.651542022-09-13T20:28:31.338867Z"Zhang, Xiaodi"https://zbmath.org/authors/?q=ai:zhang.xiaodi"Ding, Qianqian"https://zbmath.org/authors/?q=ai:ding.qianqianSummary: This paper considers charge-conservative finite element approximation and three coupled iterations of stationary inductionless magnetohydrodynamics equations in Lipschitz domain. Using a mixed finite element method, we discretize the hydrodynamic unknowns by stable velocity-pressure finite element pairs, discretize the current density and electric potential by \(\boldsymbol{H}(\operatorname{div},\varOmega)\times L^2(\varOmega)\)-comforming finite element pairs. The well-posedness of this formula and the optimal error estimate are provided. In particular, we show that the error estimates for the velocity, the current density and the pressure are independent of the electric potential. With this, we propose three coupled iterative methods: Stokes, Newton and Oseen iterations. Rigorous analysis of convergence and stability for different iterative schemes are provided, in which we improve the stability conditions for both Stokes and Newton iterative method. Numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed methods.Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domainshttps://zbmath.org/1491.651552022-09-13T20:28:31.338867Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Ochiai, Hiroyuki"https://zbmath.org/authors/?q=ai:ochiai.hiroyuki"Tanaka, Yoshitaro"https://zbmath.org/authors/?q=ai:tanaka.yoshitaroSummary: The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, \(N^2a^N\) order, where \(a\) is a positive constant less than one and \(N\) is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points \(N\).A new patch up technique for elliptic partial differential equation with irregularitieshttps://zbmath.org/1491.651562022-09-13T20:28:31.338867Z"Singh, Swarn"https://zbmath.org/authors/?q=ai:singh.swarn"Singh, Suruchi"https://zbmath.org/authors/?q=ai:singh.suruchi"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilinSummary: This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution \(u\) and the derivatives \(u_x,u_{xx}\) and the mixed derivative \(u_{xy}\). The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme.Convergence analysis of the continuous and discrete non-overlapping double sweep domain decomposition method based on PMLs for the Helmholtz equationhttps://zbmath.org/1491.651572022-09-13T20:28:31.338867Z"Kim, Seungil"https://zbmath.org/authors/?q=ai:kim.seungil"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.2|zhang.hui.8|zhang.hui.1|zhang.hui.7|zhang.hui.9|zhang.hui.5|zhang.hui.3|zhang.hui.11|zhang.hui.10|zhang.hui.6|zhang.hui.4|zhang.huiSummary: In this paper we will analyze the convergence of the non-overlapping double sweep domain decomposition method (DDM) with transmission conditions based on PMLs for the Helmholtz equation. The main goal is to establish the convergence of the double sweep DDM of both the continuous level problem and the corresponding finite element problem. We show that the double sweep process can be viewed as a contraction mapping of boundary data used for local subdomain problems not only in the continuous level and but also in the discrete level. It turns out that the contraction factor of the contraction mapping of the continuous level problem is given by an exponentially small factor determined by PML strength and PML width, whereas the counterpart of the discrete level problem is governed by the dominant term between the contraction factor similar to that of the continuous level problem and the maximal discrete reflection coefficient resulting from fast decaying evanescent modes. Based on this analysis we prove the convergence of approximate solutions in the \(H^1\)-norm. We also analyze how the discrete double sweep DDM depends on the number of subdomains and the PML parameters as the finite element discretization resolves sufficiently the Helmholtz and PML equations. Our theoretical results suggest that the contraction factor for the propagating modes depends linearly on the number of subdomains. To ensure the convergence, it is sufficient to have the PML width growing logarithmically with the number of subdomains. In the end, numerical experiments illustrating the convergence will be presented as well.A mesh-free method using piecewise deep neural network for elliptic interface problemshttps://zbmath.org/1491.651582022-09-13T20:28:31.338867Z"He, Cuiyu"https://zbmath.org/authors/?q=ai:he.cuiyu"Hu, Xiaozhe"https://zbmath.org/authors/?q=ai:hu.xiaozhe"Mu, Lin"https://zbmath.org/authors/?q=ai:mu.linThe aim in this paper is to investigate the use of deep learning methods to solve interface problems. One focuses on the second-order elliptic interface problem and the research follows the following scheme: the interface problem is reformulated by means of a least-square (LS) approach as a minimization problem and two deep neural network structures (DNN) are built to approximate the solution on sub-domains. The following second-order scalar elliptic interface problem is considered:
\[
-\nabla.(\beta(\mathcal{X})\nabla u)=f,\text{ in }\Omega_1\cup\Omega_2,
\]
\[
[[u]]=g_i\text{ on }\Gamma,
\]
\[
[[\beta(\mathcal{X})\nabla u.\mathcal N]]=g_f\text{ on }\Gamma,
\]
\[
u=g_D,\text{ on }\partial\Omega
\]
where the interface \(\Gamma\) is closed and divides domain \(\Omega\) in two disjoint sub-domains \(\Omega _1\) (the interior) and \(\Omega _2\) (the exterior). One defines the (LS) functional incorporating the interface as well as boundary conditions \(\mathcal {J}\)(v;\(g_j\),\(g_f\),\(g_D\),f) and one looks for its minimum for v\(\in\) \(H^1\)(\(\Omega\)). The construction of the two neural networks to approximate the solution on both sub-domains \(\Omega_1\) and \(\Omega _2\) is discussed in the third section of the article. One develops an adaptive deep LS algorithm. Results on numerical experiments as for instance on Sunflower Shape Interface, Sphere Shape Interface, Heart Shape Interface, Circle Interface with High Contrast Coefficients, Flower Shape Interface are presented in the fourth section of the paper.
Reviewer: Claudia Simionescu-Badea (Wien)Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equationhttps://zbmath.org/1491.651592022-09-13T20:28:31.338867Z"Cai, Jiaxiang"https://zbmath.org/authors/?q=ai:cai.jiaxiang"Chen, Juan"https://zbmath.org/authors/?q=ai:chen.juan"Chen, Min"https://zbmath.org/authors/?q=ai:chen.min.2|chen.min.1|chen.min|chen.min.3Summary: A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.Composite symmetric second derivative general linear methods for Hamiltonian systemshttps://zbmath.org/1491.651602022-09-13T20:28:31.338867Z"Talebi, Behnaz"https://zbmath.org/authors/?q=ai:talebi.behnaz"Abdi, Ali"https://zbmath.org/authors/?q=ai:abdi.ali"Hojjati, Gholamreza"https://zbmath.org/authors/?q=ai:hojjati.gholamrezaSummary: Symmetric second derivative general linear methods (SGLMs) have been already introduced for the numerical solution of time-reversible differential equations. To construct suitable high order methods for such problems, the newly developed composition theory has been successfully used for structure-preserving methods. In this paper, composite symmetric SGLMs are introduced using the generalization of composite theory for general linear methods. Then, we construct symmetric methods of order six by the composition of symmetric SGLMs of order four. Numerical results of the constructed methods verify the theoretical order of accuracy and illustrate that the invariants of motion over long time intervals for reversible Hamiltonian systems are well preserved.Computation of periodic orbits for piecewise linear oscillator by harmonic balance methodshttps://zbmath.org/1491.651612022-09-13T20:28:31.338867Z"Pei, Lijun"https://zbmath.org/authors/?q=ai:pei.lijun"Chong, Antonio S. E."https://zbmath.org/authors/?q=ai:chong.antonio-s-e"Pavlovskaia, Ekaterina"https://zbmath.org/authors/?q=ai:pavlovskaia.ekaterina-e"Wiercigroch, Marian"https://zbmath.org/authors/?q=ai:wiercigroch.marianSummary: In this paper, Harmonic Balance based methods, namely Incremental Harmonic Balance Method and the method of Harmonic Balance with Alternating Frequency and Time traditionally used to compute periodic orbits of smooth nonlinear dynamical systems, are employed to investigate the dynamics of a non-smooth system, specifically a piecewise linear oscillator with a play. The Incremental Harmonic Balance Method was used to compute the period one orbits, including those exhibiting grazing and large impacts. The method of Harmonic Balance with Alternating Frequency and Time was implemented to calculate more complex orbits and multi stability. A good agreement between obtained approximate solutions and numerically calculated responses indicates robustness of the implemented HBMs, which should allow to effectively study the global dynamics of non-smooth systems.A numerical method for the approximation of stable and unstable manifolds of microscopic simulatorshttps://zbmath.org/1491.651622022-09-13T20:28:31.338867Z"Siettos, Constantinos"https://zbmath.org/authors/?q=ai:siettos.constantinos-i"Russo, Lucia"https://zbmath.org/authors/?q=ai:russo.luciaSummary: We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.Rapid application of the spherical harmonic transform via interpolative decomposition butterfly factorizationhttps://zbmath.org/1491.651632022-09-13T20:28:31.338867Z"Bremer, James"https://zbmath.org/authors/?q=ai:bremer.james-c"Chen, Ze"https://zbmath.org/authors/?q=ai:chen.ze"Yang, Haizhao"https://zbmath.org/authors/?q=ai:yang.haizhaoConvergence of S-iterative method to a solution of Fredholm integral equation and data dependencyhttps://zbmath.org/1491.651642022-09-13T20:28:31.338867Z"Atalan, Yunus"https://zbmath.org/authors/?q=ai:atalan.yunus"Gursoy, Faik"https://zbmath.org/authors/?q=ai:gursoy.faik"Khan, Abdul Rahim"https://zbmath.org/authors/?q=ai:khan.abdul-rahimSummary: The convergence of normal S-iterative method to solution of a nonlinear Fredholm integral equation with modified argument is established. The corresponding data dependence result has also been proved. An example in support of the established results is included in our analysis.An integral equation formulation of the \(N\)-body dielectric spheres problem. II: Complexity analysishttps://zbmath.org/1491.651652022-09-13T20:28:31.338867Z"Bramas, Bérenger"https://zbmath.org/authors/?q=ai:bramas.berenger"Hassan, Muhammad"https://zbmath.org/authors/?q=ai:hassan.muhammad"Stamm, Benjamin"https://zbmath.org/authors/?q=ai:stamm.benjaminSummary: This article is the second in a series of two papers concerning the mathematical study of a boundary integral equation of the second kind that describes the interaction of \(N\) dielectric spherical particles undergoing mutual polarisation. The first article presented the numerical analysis of the Galerkin method used to solve this boundary integral equation and derived \(N\)-independent convergence rates for the induced surface charges and total electrostatic energy. The current article will focus on computational aspects of the algorithm. We provide a convergence analysis of the iterative method used to solve the underlying linear system and show that the number of liner solver iterations required to obtain a solution is independent of \(N\). Additionally, we present two linear scaling solution strategies for the computation of the approximate induced surface charges. Finally, we consider a series of numerical experiments designed to validate our theoretical results and explore the dependence of the numerical errors and computational cost of solving the underlying linear system on different system parameters.
For Part I, see [\textit{M. Hassan} and \textit{B. Stamm}, ESAIM, Math. Model. Numer. Anal. 55, 65--102 (2021; Zbl 1491.65170)].Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernelshttps://zbmath.org/1491.651662022-09-13T20:28:31.338867Z"Cai, Haotao"https://zbmath.org/authors/?q=ai:cai.haotaoSummary: The original solutions of highly oscillatory integral equations usually have rapid oscillation, which means that conventional numerical approaches used to solve these equations have poor convergence. In order to overcome this difficulty, in this paper, we propose and analyze an oscillation-preserving Legendre-Galerkin method for second kind integral equations with highly oscillatory kernels. Concretely, we first incorporate the standard approximation subspace of Legendre polynomial basis with a finite number of oscillatory functions which capture the oscillation of the exact solutions. Then, we construct an efficient numerical integration scheme, yielding a fully discrete linear system. Making use of best approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces and the compactness operator theory, we establish that the fully discrete approximate equation has a unique solution and the approximate solution reaches an optimal convergence order without the influence of the wave number. In addition, we prove that for sufficiently large wave number, the spectral condition number of the corresponding linear system is uniformly bounded. At last, we use numerical examples to demonstrate the effectiveness of the proposed method.Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equationshttps://zbmath.org/1491.651672022-09-13T20:28:31.338867Z"Cakir, Musa"https://zbmath.org/authors/?q=ai:cakir.musa"Gunes, Baransel"https://zbmath.org/authors/?q=ai:gunes.baranselSummary: In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.Existence results and numerical method for solving a fourth-order nonlinear integro-differential equationhttps://zbmath.org/1491.651682022-09-13T20:28:31.338867Z"Dang Quang Long"https://zbmath.org/authors/?q=ai:dang-quang-long."Dang Quang A"https://zbmath.org/authors/?q=ai:dang-quang-a.Summary: In this paper, we consider a boundary value problem (BVP) for a fourth-order nonlinear integro-differential equation. By reducing the problem to an operator equation, we establish the existence and uniqueness of the solution and construct a numerical method for solving it. We prove that the method is of second-order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.Solving nonlinear Volterra-Fredholm integral equations using an accurate spectral collocation methodhttps://zbmath.org/1491.651692022-09-13T20:28:31.338867Z"Hamani, Fatima"https://zbmath.org/authors/?q=ai:hamani.fatima"Rahmoune, Azedine"https://zbmath.org/authors/?q=ai:rahmoune.azedineSummary: In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both \(L^\infty\) and weighted \(L^2\) norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.An integral equation formulation of the \(N\)-body dielectric spheres problem. I: Numerical analysishttps://zbmath.org/1491.651702022-09-13T20:28:31.338867Z"Hassan, Muhammad"https://zbmath.org/authors/?q=ai:hassan.muhammad"Stamm, Benjamin"https://zbmath.org/authors/?q=ai:stamm.benjaminSummary: In this article, we analyse an integral equation of the second kind that represents the solution of \(N\) interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants -- and thus the approximation error -- with respect to the number \(N\) of involved dielectric spheres. We develop a new \textit{a priori} error analysis that demonstrates \(N\)-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of \(N\).
For Part II, see [\textit{B. Bramas} et al., ESAIM, Math. Model. Numer. Anal. 55, 625--651 (2021; Zbl 1491.65165)].Numerical solution of two-dimensional weakly singular Volterra integral equations with non-smooth solutionshttps://zbmath.org/1491.651712022-09-13T20:28:31.338867Z"Katani, R."https://zbmath.org/authors/?q=ai:katani.roghayeh"McKee, S."https://zbmath.org/authors/?q=ai:mckee.stephen-k|mckee.sally-a|mckee.seanSummary: Various numerical methods have been proposed for the solution of weakly singular Volterra integral equations but, for the most part, authors have dealt with linear or one-dimensional weakly singular Volterra integral equations, or have assumed that these equations have smooth solutions. The main purpose of this paper is to propose and analyse a numerical method for the solution of two-dimensional nonlinear weakly singular Volterra integral equations of the second kind. In general the solutions of these equations exhibit singularities in their derivatives at \(t = 0\) even if the forcing functions are smooth. To overcome these difficulties a simple smoothing change of variables is proposed. By applying this transformation an equation is obtained which, while still being weakly singular, can have a solution as smooth as is required. We then solve this transformed integral equation using Navot's quadrature rule for computing integrals with an end point singularity. A new extension of a discrete Gronwall inequality allows us to prove convergence and obtain an error estimate. The theoretical results are then verified by numerical examples.Calculation of the gradient of Tikhonov's functional in solving coefficient inverse problems for linear partial differential equationshttps://zbmath.org/1491.651722022-09-13T20:28:31.338867Z"Leonov, Alexander S."https://zbmath.org/authors/?q=ai:leonov.alexander-s"Sharov, Alexander N."https://zbmath.org/authors/?q=ai:sharov.alexander-n"Yagola, Anatoly G."https://zbmath.org/authors/?q=ai:yagola.anatolii-grigorevichSummary: A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young's modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.Central part interpolation schemes for a class of fractional initial value problemshttps://zbmath.org/1491.651732022-09-13T20:28:31.338867Z"Lillemäe, Margus"https://zbmath.org/authors/?q=ai:lillemae.margus"Pedas, Arvet"https://zbmath.org/authors/?q=ai:pedas.arvet"Vikerpuur, Mikk"https://zbmath.org/authors/?q=ai:vikerpuur.mikkSummary: We consider an initial value problem for linear fractional integro-differential equations with weakly singular kernels. Using an integral equation reformulation of the underlying problem, a collocation method based on the central part interpolation by continuous piecewise polynomials on the uniform grid is constructed and analysed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.The solution of the nonlinear mixed partial integro-differential equation via two-dimensional hybrid functionshttps://zbmath.org/1491.651742022-09-13T20:28:31.338867Z"Rostami, Yaser"https://zbmath.org/authors/?q=ai:rostami.yaser"Maleknejad, Khosrow"https://zbmath.org/authors/?q=ai:maleknejad.khosrowSummary: In the present paper, a new method is introduced for the approximate solution of two-dimensional nonlinear mixed Volterra-Fredholm partial integro-differential equations with initial conditions using two-dimensional hybrid Taylor polynomials and Block-Pulse functions. For this purpose, operational metrics of product and integration of the cross-product and derivative are introduced that essentially of hybrid functions. The use of these operational matrices simplifies considerably the structure of the computation used for a set of algebraic equations for the solution of partial integro-differential equations. Convergence analysis and some numerical results are presented to illustrate the effectiveness and accuracy of the method. Some figures are plotted to demonstrate the error analysis of the proposed scheme. Use operational matrices of two-dimensional hybrid Taylor polynomials and Block-Pulse functions and obtain high accuracy of the method.Super-resolution of positive sources on an arbitrarily fine gridhttps://zbmath.org/1491.651752022-09-13T20:28:31.338867Z"Morgenshtern, Veniamin I."https://zbmath.org/authors/?q=ai:morgenshtern.veniamin-iSummary: In super-resolution it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at low frequencies capped by \(f_{\mathrm{lo}}\). In the case when the point sources are positive and are located on a grid, it has been recently established that the super-resolution problem can be solved via linear programming in a stable manner and that the method is nearly optimal in the minimax sense. The quality of the reconstruction critically depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within an interval of side length about \(1/f_{\mathrm{lo}}\). This work extends the earlier result and shows that the conclusion continues to hold when the locations of the point sources are arbitrary, i.e., the grid is arbitrarily fine. The proof relies on new interpolation constructions in Fourier analysis.A unified FFT-based approach to maximum assignment problems related to transitive finite group actionshttps://zbmath.org/1491.651762022-09-13T20:28:31.338867Z"Clausen, Michael"https://zbmath.org/authors/?q=ai:clausen.michaelSummary: This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The ``simplest'' nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach [\textit{R. Kondor}, in: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, TX, USA, January 17--19, 2010. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); New York, NY: Association for Computing Machinery (ACM). 1017--1028 (2010; Zbl 1288.68130)] to the exact solution of the asymmetric maximum quadratic assignment problem.A sample efficient sparse FFT for arbitrary frequency candidate sets in high dimensionshttps://zbmath.org/1491.651772022-09-13T20:28:31.338867Z"Kämmerer, Lutz"https://zbmath.org/authors/?q=ai:kammerer.lutz"Krahmer, Felix"https://zbmath.org/authors/?q=ai:krahmer.felix"Volkmer, Toni"https://zbmath.org/authors/?q=ai:volkmer.toniSummary: In this paper, a sublinear time algorithm is presented for the reconstruction of functions that can be represented by just few out of a potentially large candidate set of Fourier basis functions in high spatial dimensions, a so-called high-dimensional sparse fast Fourier transform. In contrast to many other such algorithms, our method works for arbitrary candidate sets and does not make additional structural assumptions on the candidate set. Our transform significantly improves upon the other approaches available for such a general framework in terms of the scaling of the sample complexity. Our algorithm is based on sampling the function along multiple rank-1 lattices with random generators. Combined with a dimension-incremental approach, our method yields a sparse Fourier transform whose computational complexity only grows mildly in the dimension and can hence be efficiently computed even in high dimensions. Our theoretical analysis establishes that any Fourier \(s\)-sparse function can be accurately reconstructed with high probability. This guarantee is complemented by several numerical tests demonstrating the high efficiency and versatile applicability for the exactly sparse case and also for the compressible case.Fast discrete Fourier transform on local fields of zero characteristichttps://zbmath.org/1491.651782022-09-13T20:28:31.338867Z"Lukomskii, S. F."https://zbmath.org/authors/?q=ai:lukomskii.sergei-fedorovich"Vodolazov, A. M."https://zbmath.org/authors/?q=ai:vodolazov.a-mA local field is locally compact, non-discrete, totally disconnected complete topological space with continuous operations of addition \((\dot{+})\) and multiplication \((\cdot)\), which satisfy the field axioms. The local field \(K\) of zero characteristic is a finite extension of the degree \(n=se\) of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers, where \(p\) is a prime number and depends on two parameters \(s, e\in\mathbb{N}\). The Rademacher function is defined as the character \(r_m\in G^{\bot}_{m+1} \backslash G^{\bot}_{m}\) in a zero-dimensional group \(G\) with a basic chain of subgroups \((G_m)\).
Let \(\overline{\alpha}_m=(\alpha_{m,0}, \alpha_{m,1},\ldots, \alpha_{m,s-1})\in GF(p^s)\). Then the generalized Walsh functions are defined as \(\mathbf{r}_m^{\overline{\alpha}_m}:=r_{ms+0}^{\alpha_{m,0}} r_{ms+1}^{\alpha_{m,1}} \ldots r_{ms+s-1}^{\alpha_{m,s-1}}\), where \(r_{ms+j}, \ 0\le j< s, m\in\mathbb{Z}\) are Rademacher functions. Here, \(\mathfrak{D}_{K_{(N)}}(K_{(-M)})\) denotes the set of functions \(f\in L_2(K)\) such that (i) supp \(f\subset K_{(-M)}\), and (ii) \(f\) is constant on cosets \(K_{(N)}\dot{+} g\) holds. Each function \(f^{(N+1)}\in \mathfrak{D}_{K_{(N+1)}}(K_{(0)})\) can be represented by a finite sum \(f^{(N+1)}=\sum\limits_{\overline{\alpha}_0,\ldots,\overline{\alpha}_N \in GF(p^s)}c_{\overline{\alpha}_0\ldots\overline{\alpha}_N}\mathbf{r}^{\overline{\alpha}_0}_0\ldots \mathbf{r}^{\overline{\alpha}_N}_N\).
In this paper, the authors obtain the explicit form of Rademacher functions in the field \(K\) and describe a representation of the characters of the field \(K\) in terms of Rademacher functions. They obtain a fast algorithm for calculating the coefficients \(c_{\overline{\alpha}_0\ldots\overline{\alpha}_N}\) and thus give the calculation formulas for the inverse and direct Fourier transforms in the field \(K\). The total number of operations required to complete the algorithm is found to be \((s+1)p^s \frac{N^2}{e}p^{s(N+1)}\) which gives the complexity of both direct and inverse transforms as \(M \log^2 M\) which can be further reduced to \(M \log M\).
Reviewer: Nikhil Khanna (Masqaṭ)On the use of the infinity computer architecture to set up a dynamic precision floating-point arithmetichttps://zbmath.org/1491.651792022-09-13T20:28:31.338867Z"Amodio, Pierluigi"https://zbmath.org/authors/?q=ai:amodio.pierluigi"Brugnano, Luigi"https://zbmath.org/authors/?q=ai:brugnano.luigi"Iavernaro, Felice"https://zbmath.org/authors/?q=ai:iavernaro.felice"Mazzia, Francesca"https://zbmath.org/authors/?q=ai:mazzia.francescaSummary: We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix \textcircled{1}. The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.
Editorial remark: For more information on the notion of grossone, introduced by \textit{Y. D. Sergeyev}, see [Arithmetic of infinity. Cosenza: Edizioni Orizzonti Meridionali (2003; Zbl 1076.03048); EMS Surv. Math. Sci. 4, No. 2, 219--320 (2017; Zbl 1390.03048)]; see also [\textit{A. E. Gutman} and \textit{S. S. Kutateladze}, Sib. Mat. Zh. 49, No. 5, 1054--1076 (2008; Zbl 1224.03045); translation in Sib. Math. J. 49, No. 5, 835--841 (2008)].Euro-Par 2021: parallel processing workshops. Euro-Par 2021 international workshops, Lisbon, Portugal, August 30--31, 2021. Revised selected papershttps://zbmath.org/1491.680142022-09-13T20:28:31.338867ZThe articles of this volume will not be indexed individually. For the conference proceedings of Euro-Par 2021 see [Zbl 1483.68013].Deep learning based classification of time series of Chen and Rössler chaotic systems over their graphic imageshttps://zbmath.org/1491.681772022-09-13T20:28:31.338867Z"Aricioğlu, Burak"https://zbmath.org/authors/?q=ai:aricioglu.burak"Uzun, Süleyman"https://zbmath.org/authors/?q=ai:uzun.suleyman"Kaçar, Sezgin"https://zbmath.org/authors/?q=ai:kacar.sezginSummary: In this study, the graphic images of time series of different chaotic systems are classified with deep learning methods for the first time in the literature. For the classification, a dataset contains images of time series of Chen and Rössler chaotic systems for different parameter values, initial conditions, step size and time length are generated. Then, high accuracy classifications are performed with transfer learning methods. The used transfer learning methods are \textit{SqueezeNet}, \textit{VGG-19}, \textit{AlexNet}, \textit{ResNet50}, \textit{ResNet-101}, \textit{DenseNet-201}, \textit{ShuffleNet}, and \textit{GoogLeNet}. According to the problem, classifications accuracy is varying between 89\% and 99.7\% in this study. Thus, this study shows that identifying a chaotic system from its graphic image of time series is possible.CMD: controllable matrix decomposition with global optimization for deep neural network compressionhttps://zbmath.org/1491.681862022-09-13T20:28:31.338867Z"Zhang, Haonan"https://zbmath.org/authors/?q=ai:zhang.haonan"Liu, Longjun"https://zbmath.org/authors/?q=ai:liu.longjun"Zhou, Hengyi"https://zbmath.org/authors/?q=ai:zhou.hengyi"Sun, Hongbin"https://zbmath.org/authors/?q=ai:sun.hongbin.1"Zheng, Nanning"https://zbmath.org/authors/?q=ai:zheng.nanningSummary: The compression and acceleration of Deep neural networks (DNNs) are necessary steps to deploy sophisticated networks into resource-constrained hardware systems. Due to the weight matrix tends to be low-rank and sparse, several low-rank and sparse compression schemes are leveraged to reduce the overwhelmed weight parameters of DNNs. In these previous schemes, how to make the most of the low-rank and sparse components of weight matrices and how to globally decompose the weight matrix of different layers for efficient compression need to be further investigated. In this paper, in order to effectively utilize the low-rank and sparse characteristics of the weight matrix, we first introduce a sparse coefficient to dynamically control the allocation between the low-rank and sparse components, and an efficient reconstructed network is designed to reduce the inference time. Secondly, since the results of low-rank decomposition can affect the compression ratio and accuracy of DNNs, we establish an optimization problem to automatically select the optimal hyperparameters of the compressed network and achieve global compression for all the layers of network synchronously. Finally, to solve the optimization problem, we present a decomposition-searching algorithm to search the optimal solution. The algorithm can dynamically keep the balance between the compression ratio and accuracy. Extensive experiments of AlexNet, VGG-16 and ResNet-18 on CIFAR-10 and ImageNet are employed to evaluate the effectiveness of the proposed approach. After slight fine-tuning, compressed networks have gained \(1.2\times\) to \(11.3\times\) speedup and our method reduces the size of different networks by \(1.4\times\) to \(14.6\times \).A unified B-spline framework for scale-invariant keypoint detectionhttps://zbmath.org/1491.682512022-09-13T20:28:31.338867Z"Zheng, Qi"https://zbmath.org/authors/?q=ai:zheng.qi"Gong, Mingming"https://zbmath.org/authors/?q=ai:gong.mingming"You, Xinge"https://zbmath.org/authors/?q=ai:you.xinge"Tao, Dacheng"https://zbmath.org/authors/?q=ai:tao.dachengSummary: Scale-invariant keypoint detection is a fundamental problem in low-level vision. To accelerate keypoint detectors (e.g. DoG, Harris-Laplace, Hessian-Laplace) that are developed in Gaussian scale-space, various fast detectors (e.g., SURF, CenSurE, and BRISK) have been developed by approximating Gaussian filters with simple box filters. However, there is no principled way to design the shape and scale of the box filters. Additionally, the involved integral image technique makes it difficult to figure out the continuous kernels that correspond to the discrete ones used in these detectors, so there is no guarantee that those good properties such as causality in the original Gaussian space can be inherited. To address these issues, in this paper, we propose a unified B-spline framework for scale-invariant keypoint detection. Owing to an approximate relationship to Gaussian kernels, the B-spline framework provides a mathematical interpretation of existing fast detectors based on integral images. In addition, from B-spline theories, we illustrate the problem in repeated integration, which is the generalized version of the integral image technique. Finally, following the dominant measures for keypoint detection and automatic scale selection, we develop B-spline determinant of Hessian (B-DoH) and B-spline Laplacian-of-Gaussian (B-LoG) as two instantiations within the unified B-spline framework. For efficient computation, we propose to use \textit{repeated running-sums} to convolve images with B-spline kernels with fixed orders, which avoids the problem of integral images by introducing an extra interpolation kernel. Our B-spline detectors can be designed in a principled way without the heuristic choice of kernel shape and scales and naturally extend the popular SURF and CenSurE detectors with more complex kernels. Extensive experiments on the benchmark dataset demonstrate that the proposed detectors outperform the others in terms of repeatability and efficiency.Inextensible surface reconstruction under small relative deformations from distributed angle measurementshttps://zbmath.org/1491.682602022-09-13T20:28:31.338867Z"Talon, Thibaud"https://zbmath.org/authors/?q=ai:talon.thibaud"Pellegrino, Sergio"https://zbmath.org/authors/?q=ai:pellegrino.sergioSummary: A mathematical model to measure the shape of a 3D surface using angle measurements from embedded sensors is presented. The surface is known in a reference configuration and is assumed to have deformed inextensibly to its current shape. An inextensibility condition is enforced through a discretization of the metric tensor generating a finite number of constraints. This model allows to parameterize the shape of the surface using a small number of unknowns which leads to a small number of sensors. We study the singularities of the equations and derive necessary conditions for the problem to be well-posed as well as limitations of the algorithm. Simulations and experiments are performed on developable surfaces under relatively small deformations to analyze the performance of the method and to show the influence of the parameters used in our algorithm. Overall, the proposed method outperforms the current state-of-the-art by almost an order of magnitude.Formal correctness of comparison algorithms between binary64 and decimal64 floating-point numbershttps://zbmath.org/1491.682632022-09-13T20:28:31.338867Z"Blot, Arthur"https://zbmath.org/authors/?q=ai:blot.arthur"Muller, Jean-Michel"https://zbmath.org/authors/?q=ai:muller.jean-michel"Théry, Laurent"https://zbmath.org/authors/?q=ai:thery.laurentSummary: We present a full \textsc{Coq} formalisation of the correctness of some comparison algorithms between binary64 and decimal64 floating-point numbers, using computation intensive proofs and a continued fractions library built for this formalisation.
For the entire collection see [Zbl 1369.68006].Free vibration analysis of combined composite laminated conical-cylindrical shells with varying thickness using the Haar wavelet methodhttps://zbmath.org/1491.740392022-09-13T20:28:31.338867Z"Kim, Kwanghun"https://zbmath.org/authors/?q=ai:kim.kwanghun"Kwak, Songhun"https://zbmath.org/authors/?q=ai:kwak.songhun"Pang, Cholho"https://zbmath.org/authors/?q=ai:pang.cholho"Pang, Kyongjin"https://zbmath.org/authors/?q=ai:pang.kyongjin"Choe, Kwangil"https://zbmath.org/authors/?q=ai:choe.kwangilSummary: This paper presents the free vibration analysis of combined composite laminated conical-cylindrical shells with varying thickness using the Haar wavelet method (HWM). The displacement field of the combined shell is set based on the first-order shear deformation theory (FSDT), the displacement components, and rotation of individual shells including boundary conditions that are expanded by the Haar wavelet and Fourier series in the meridional and the circumferential direction. By solving the vibration characteristic equation discretized by the Haar wavelet, the vibrational results of combined shells are obtained. Then, the results of the proposed method are compared with those of published literature and finite element analysis (FEA). The results show that HWM has high convergence and high accuracy for the free vibration analysis of the combined composite laminated conical-cylindrical shells with varying thickness. Also, the effects of the parameters such as thickness variation parameters, material properties, geometrical dimensions, and different boundary conditions, on the vibrational behavior of the combined shells are investigated. Finally, new numerical results are provided to illustrate the free vibration behavior of the combined composite laminated conical-cylindrical shells with varying thickness.Structure-preserving discretization of a coupled heat-wave system, as interconnected port-Hamiltonian systemshttps://zbmath.org/1491.760442022-09-13T20:28:31.338867Z"Haine, Ghislain"https://zbmath.org/authors/?q=ai:haine.ghislain"Matignon, Denis"https://zbmath.org/authors/?q=ai:matignon.denisSummary: The heat-wave system is recast as the coupling of port-Hamiltonian subsystems (pHs), and discretized in a structure-preserving way by the partitioned finite element method (PFEM) [\textit{F. L. Cardoso-Ribeiro} et al., IMA J. Math. Control Inf. 38, No. 2, 493--533 (2021; Zbl 1475.93051); ``A structure-preserving Partitioned Finite Element Method for the 2D wave equation'', IFAC-PapersOnLine 51, No. 3, 119--124 (2018; \url{doi:10.1016/j.ifacol.2018.06.033})]. Then, depending on the geometric configuration of the two domains, different asymptotic behaviours of the energy of the coupled system can be recovered at the numerical level, assessing the validity of the theoretical results of \textit{X. Zhang} and \textit{E. Zuazua} [Arch. Ration. Mech. Anal. 184, No. 1, 49--120 (2007; Zbl 1178.74075)].
For the entire collection see [Zbl 1482.94007].The macroelement analysis for axisymmetric Stokes equationshttps://zbmath.org/1491.760452022-09-13T20:28:31.338867Z"Lee, Young-Ju"https://zbmath.org/authors/?q=ai:lee.youngju"Li, Hengguang"https://zbmath.org/authors/?q=ai:li.hengguangSummary: We consider the mixed finite element approximation of the axisymmetric Stokes problem (ASP) on a bounded polygonal domain in the \(rz\)-plane. Standard stability results on mixed methods do not apply due to the singular coefficients in the differential operator and due to the singular or vanishing weights in the associated function spaces. We develop new finite element analysis in these weighted spaces, and propose macroelement conditions that are sufficient to ensure the well-posedness of the mixed methods for the ASP. These conditions are local, relatively easy to verify, and therefore will be useful for validating the stability of a variety of mixed finite element methods. These new conditions can not only re-verify existing stable mixed methods for the ASP, but also lead to the discovery of new stable conservative mixed methods. We report numerical test results that confirm the theory.A fast convergent semi-analytic method for an electrohydrodynamic flow in a circular cylindrical conduithttps://zbmath.org/1491.760532022-09-13T20:28:31.338867Z"Abukhaled, Marwan"https://zbmath.org/authors/?q=ai:abukhaled.marwan-i"Khuri, S. A."https://zbmath.org/authors/?q=ai:khuri.suheil-aSummary: A semi-analytical solution of the nonlinear boundary value problem that models the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit is presented. An integral operator expressed in terms of Green's function is constructed then followed by an application of fixed point theory to generate a highly accurate semi-analytical expression of the fluid velocity for all possible values of relevant parameters. A proof of convergence for the proposed method, based on the contraction mapping principle, is presented. Numerical simulations and comparison with other analytical methods confirm that the proposed approach is convergent, stable, and highly accurate.Multisymplectic variational integrators for fluid models with constraintshttps://zbmath.org/1491.760542022-09-13T20:28:31.338867Z"Demoures, François"https://zbmath.org/authors/?q=ai:demoures.francois"Gay-Balmaz, François"https://zbmath.org/authors/?q=ai:gay-balmaz.francoisSummary: We present a structure preserving discretization of the fundamental spacetime geometric structures of fluid mechanics in the Lagrangian description in 2D and 3D. Based on this, multisymplectic variational integrators are developed for barotropic and incompressible fluid models, which satisfy a discrete version of Noether theorem. We show how the geometric integrator can handle regular fluid motion in vacuum with free boundaries and constraints such as the impact against an obstacle of a fluid flowing on a surface. Our approach is applicable to a wide range of models including the Boussinesq and shallow water models, by appropriate choice of the Lagrangian.
For the entire collection see [Zbl 1482.94007].Metriplectic integrators for dissipative fluidshttps://zbmath.org/1491.760552022-09-13T20:28:31.338867Z"Kraus, Michael"https://zbmath.org/authors/?q=ai:kraus.michaelSummary: Many systems from fluid dynamics and plasma physics possess a so-called metriplectic structure, that is the equations are comprised of a conservative, Hamiltonian part, and a dissipative, metric part. Consequences of this structure are conservation of important quantities, such as mass, momentum and energy, and compatibility with the laws of thermodynamics, e.g., monotonic dissipation of entropy and existence of a unique equilibrium state.
For simulations of such systems to deliver accurate and physically correct results, it is important to preserve these relations and conservation laws in the course of discretisation. This can be achieved most easily not by enforcing these properties directly, but by preserving the underlying abstract mathematical structure of the equations, namely their metriplectic structure. From that, the conservation of the aforementioned desirable properties follows automatically.
This paper describes a general and flexible framework for the construction of such metriplectic structure-preserving integrators, that facilitates the design of novel numerical methods for systems from fluid dynamics and plasma physics.
For the entire collection see [Zbl 1482.94007].An adaptive virtual element method for incompressible flowhttps://zbmath.org/1491.760572022-09-13T20:28:31.338867Z"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying|wang.ying.2|wang.ying.8|wang.ying.4|wang.ying.5|wang.ying.6|wang.ying.3|wang.ying.1"Wang, Gang"https://zbmath.org/authors/?q=ai:wang.gang.5|wang.gang.4|wang.gang.1|wang.gang.3|wang.gang|wang.gang.2"Wang, Feng"https://zbmath.org/authors/?q=ai:wang.feng.3|wang.feng.4|wang.feng.1|wang.feng.2Summary: In this paper, we firstly present and analyze a residual-type a posteriori error estimator for a low-order virtual element discretization for the Stokes problem on general polygonal meshes. We prove that this estimator yields globally upper and locally lower bounds for the discretization error. Then, we extend the estimator to the Navier-Stokes problem. In order to deal with the case of small viscosity, we modify the discrete bilinear form following the idea of variational multiscale method. Since the virtual element method naturally handles hanging nodes, the mesh refinement can exploit them without any local refinement to recover mesh conformity. A series of benchmark tests are reported to verify the effectiveness and flexibility of the designed error estimator when it is combined with adaptive mesh refinement.Accuracy of a low Mach number model for time-harmonic acousticshttps://zbmath.org/1491.760642022-09-13T20:28:31.338867Z"Mercier, J.-F."https://zbmath.org/authors/?q=ai:mercier.jean-francoisA method to enhance the noise robustness of correlation velocity measurement using discrete wavelet transformhttps://zbmath.org/1491.760652022-09-13T20:28:31.338867Z"Son, Pong-Chol"https://zbmath.org/authors/?q=ai:son.pong-chol"Kim, Kyong-Il"https://zbmath.org/authors/?q=ai:kim.kyong-il"Choe, Kyong-Chol"https://zbmath.org/authors/?q=ai:choe.kyong-chol"Kye, Hyok-Il"https://zbmath.org/authors/?q=ai:kye.hyok-ilCorrelation velocity measurement techniques are efficiently used to measure the velocity of underwater vehicles, and the maximum correlation coefficient is an important parameter of correlation velocity measurement. In this paper, the relationship between the signal-to-noise ratio (SNR) of the received signal and the maximum value of the correlation matrix is considered, and the maximum correlation coefficient equation according to SNR is presented. Especially, a wavelet thresholding denoising method is successfully used to improve SNR so that the noise robustness of correlation velocity measurement is proposed, and the modified maximum correlation coefficient equation according to SNR is provided accordingly. Simulation results show that the new method of correlation velocity measurement using wavelet thresholding could get the largest maximum correlation coefficients according to SNR steadily compared with the classical method. In particular, the performance improvements in correlation velocity log (CVL) operating under low SNR below 6 dB are more significant.
Reviewer: Yankui Sun (Beijing)Fourier theory in optics and optical information processinghttps://zbmath.org/1491.780012022-09-13T20:28:31.338867Z"Yatagai, Toyohiko"https://zbmath.org/authors/?q=ai:yatagai.toyohikoPublisher's description: Fourier analysis is one of the most important concepts when you apply physical ideas to engineering issues. This book provides a comprehensive understanding of Fourier transform and spectral analysis in optics, image processing, and signal processing. Written by a world renowned author, this book looks to unify the readers understanding of principles of optics, information processing and measurement. This book describes optical imaging systems through a linear system theory. The book also provides an easy understanding of Fourier transform and system theory in optics. It also provides background of optical measurement and signal processing. Finally, the author also provides a systematic approach to learning many signal processing techniques in optics. The book is intended for researchers, industry professionals, and graduate level students in optics and information processing.Nine equilibrium points of four point charges on the planehttps://zbmath.org/1491.780022022-09-13T20:28:31.338867Z"Lee, Tsung-Lin"https://zbmath.org/authors/?q=ai:lee.tsung-lin"Tsai, Ya-Lun"https://zbmath.org/authors/?q=ai:tsai.ya-lunSummary: We find a specific configuration for four point charges on the plane and show, with charge values in a small region, there are nine equilibrium points on the same plane, which reaches the claimed upper bound of Maxwell's conjecture. A procedure of computing bifurcation curves is presented for assisting in locating the region in the parameter space yielding the nine equilibrium points.Hybridization of the rigorous coupled-wave approach with transformation optics for electromagnetic scattering by a surface-relief gratinghttps://zbmath.org/1491.780092022-09-13T20:28:31.338867Z"Civiletti, B. J."https://zbmath.org/authors/?q=ai:civiletti.benjamin-j"Lakhtakia, A."https://zbmath.org/authors/?q=ai:lakhtakia.akhlesh"Monk, P. B."https://zbmath.org/authors/?q=ai:monk.peter-bThe authors combine transformation optics with the rigorous coupled-wave approach, in view to study the time-harmonic Maxwell equations in a spatial domain that contains a grating, being invariant in one dimension (and so that the chosen constitutive properties allow the reduction of the full Maxwell system to a 2D Helmholtz equation for each linear polarization state). The existence of solution to the original scattering problem is obtained. A convergence analysis was included for a discretized form of the transformed problem (with respect to two different parameters), and the uniqueness of solution of this discretized problem is obtained. A numerical example was presented as a test of the convergence theory, allowing also some comparison with other known methods.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Effects of electric field on multiple vibrational resonances in Hindmarsh-Rose neuronal systemshttps://zbmath.org/1491.780162022-09-13T20:28:31.338867Z"Wang, Guowei"https://zbmath.org/authors/?q=ai:wang.guowei"Yu, Dong"https://zbmath.org/authors/?q=ai:yu.dong"Ding, Qianming"https://zbmath.org/authors/?q=ai:ding.qianming"Li, Tianyu"https://zbmath.org/authors/?q=ai:li.tianyu"Jia, Ya"https://zbmath.org/authors/?q=ai:jia.yaSummary: The effects of electric field on vibrational resonance in single Hindmarsh-Rose (HR) neuron and coupled HR neurons system are investigated by using Fourier coefficient, respectively. It is found that the multiple vibrational resonances (MVR) can be observed in a single HR neuron model no matter the electric field is considered or not, and the electric field weakens the MVR. When bidirectional coupling between two HR neurons is considered, the occurrence of MVR can also be detected, it is very interesting to observe that the electric field can enhance the MVR. The higher the frequency of the low-frequency signal is, the less the number of resonance peaks of the system response to the low-frequency signal will be. Moreover, the local anti-resonance is also observed when appropriate parameters are selected. The effects of coupling strength and other system parameters on Fourier coefficient are also illustrated here. The systems manifesting MVR have better capacity for detecting and propagating signals.FXRS: fast X-ray spectrum-simulator theory and software implementationhttps://zbmath.org/1491.780172022-09-13T20:28:31.338867Z"Chirilǎ, Ciprian C."https://zbmath.org/authors/?q=ai:chirila.ciprian-c"Ha, T. M. H."https://zbmath.org/authors/?q=ai:ha.t-m-hSummary: We propose a simple, computationally efficient scheme for an X-ray spectrum simulator. The theoretical models describing the physical processes involved are employed in our Monte Carlo software in a coherent way, paving the way for straightforward future improvements. Our results compare satisfactorily to experimental results from literature and to results from dedicated simulation software. The simplicity, excellent statistical errors, and short execution time of our code recommend it for intensive use in X-ray generation simulations.Model reduction in computational homogenization for transient heat conductionhttps://zbmath.org/1491.800052022-09-13T20:28:31.338867Z"Waseem, A."https://zbmath.org/authors/?q=ai:waseem.abdullah"Heuzé, T."https://zbmath.org/authors/?q=ai:heuze.thomas"Stainier, L."https://zbmath.org/authors/?q=ai:stainier.laurent"Geers, M. G. D."https://zbmath.org/authors/?q=ai:geers.marc-g-d"Kouznetsova, V. G."https://zbmath.org/authors/?q=ai:kouznetsova.varvara-gThe authors propose a computationally efficient homogenization method for transient heat conduction problems posed in non-homogeneous media. They consider the transient heat problem (energy balance) \(\overline{\nabla } \cdot \overline{q}+\overline{\overset{.}{\varepsilon }}=0\) where \(\overline{q }\) and \(\overline{\overset{.}{\varepsilon }}\) are the macroscopic heat flow and the rate of change of macroscopic internal energy.\ At a microscale, the problem is written as \(\nabla \cdot q+\overset{.}{\varepsilon }=0\) and Fourier's law \(q=-\lambda \nabla \theta \) is added. These equations may be completed with boundary conditions. The authors then present the downscaling and upscaling process to transfer information concerning the temperature between the two scales. They introduce a model order reduction using a finite element formulation, first considering tied and retained nodes, writing a microscale discrete problem and decomposing the microscopic solution in a steady-state and a transient components. They here assume a linearity hypothesis and a relaxed separation of scales. This allows to derive the macroscopic quantities through an averaging process. In the last part of their paper, the authors apply their method in the case of a microstructure containing randomly distributed circular inclusions. They present the steady-state and transient reduced bases and the results they obtain together with the computational cost. The paper ends with a presentation of the applicability limits of this method in the case where significant transient phenomena occur.
Reviewer: Alain Brillard (Riedisheim)Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approachhttps://zbmath.org/1491.820142022-09-13T20:28:31.338867Z"Breiten, Tobias"https://zbmath.org/authors/?q=ai:breiten.tobias"Hartmann, Carsten"https://zbmath.org/authors/?q=ai:hartmann.carsten"Neureither, Lara"https://zbmath.org/authors/?q=ai:neureither.lara"Sharma, Upanshu"https://zbmath.org/authors/?q=ai:sharma.upanshuSummary: We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow [\textit{Y. Chen} et al., J. Math. Phys. 56, No. 11, 113302, 17 p. (2015; Zbl 1327.82063)] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower (``target'') or the higher (``simulation'') temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observations from the perspective of stochastic optimization algorithms and enhanced sampling schemes in molecular dynamics.
{\copyright 2021 American Institute of Physics}On a class of Fokker-Planck equations with subcritical confinementhttps://zbmath.org/1491.820162022-09-13T20:28:31.338867Z"Toscani, Giuseppe"https://zbmath.org/authors/?q=ai:toscani.giuseppe"Zanella, Mattia"https://zbmath.org/authors/?q=ai:zanella.mattiaSummary: We study the relaxation to equilibrium for a class of linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for any given probability density \(e(x)\), the diffusion coefficient can be built to have \(e(x)\) as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density \(e(x)\), a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.Symmetric scalarshttps://zbmath.org/1491.830212022-09-13T20:28:31.338867Z"Grall, Tanguy"https://zbmath.org/authors/?q=ai:grall.tanguy"Jazayeri, Sadra"https://zbmath.org/authors/?q=ai:jazayeri.sadra"Pajer, Enrico"https://zbmath.org/authors/?q=ai:pajer.enrico(no abstract)Spontaneous radiation of black holeshttps://zbmath.org/1491.830362022-09-13T20:28:31.338867Z"Zeng, Ding-fang"https://zbmath.org/authors/?q=ai:zeng.ding-fangSummary: We provide an explicitly hermitian hamiltonian description for the spontaneous radiation of black holes, which is a many-level, multiple-degeneracy generalization of the usual Janeys-Cummings model for two-level atoms. We show that under single-particle radiation and standard Wigner-Wiesskopf approximation, our model yields exactly thermal type power spectrum as hawking radiation requires. While in the many-particle radiation cases, numeric methods allow us to follow the evolution of microscopic state of a black hole exactly, from which we can get the firstly increasing then decreasing entropy variation trend for the radiation particles just as the Page-curve exhibited. Basing on this model analysis, we claim that two ingredients are necessary for resolutions of the information missing puzzle, a spontaneous radiation like mechanism for the production of hawking particles and proper account of the macroscopic superposition happening in the full quantum description of a black hole radiation evolution and, the working logic of replica wormholes is an effect account of this latter ingredient.
As the basis for our interpretation of black hole Hawking radiation as their spontaneous radiation, we also provide a fully atomic like inner structure models for their microscopic states definition and origins of their Bekenstein-Hawking entropy, that is, exact solution families to the Einstein equation sourced by matter constituents oscillating across the central point and their quantization. Such a first quantization model for black holes' microscopic state is non necessary for our spontaneous radiation description, but has advantages comparing with other alternatives, such as string theory fuzzball or brick wall models.Cosmic microwave background anisotropy numerical solution (CMBAns). I: An introduction to \(C_l\) calculationhttps://zbmath.org/1491.830592022-09-13T20:28:31.338867Z"Das, Santanu"https://zbmath.org/authors/?q=ai:das.santanu-kumar"Phan, Anh"https://zbmath.org/authors/?q=ai:phan.anh-dung|phan.anh-vu|phan.anh-huy(no abstract)Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansionshttps://zbmath.org/1491.900342022-09-13T20:28:31.338867Z"Bachi, Katia"https://zbmath.org/authors/?q=ai:bachi.katia"Chauvière, Cédric"https://zbmath.org/authors/?q=ai:chauviere.cedric"Djellout, Hacène"https://zbmath.org/authors/?q=ai:djellout.hacene"Abbas, Karim"https://zbmath.org/authors/?q=ai:abbas.karimSummary: In this paper, we develop a numerical approach based on chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modeling the propagation of uncertainty arising from those parameters. Furthermore, polynomial chaos expansion affords a natural framework for computing Sobol' indices. Such indices give reliable information on the relative importance of each uncertain entry parameters.Fair packing and covering on a relative scalehttps://zbmath.org/1491.900992022-09-13T20:28:31.338867Z"Diakonikolas, Jelena"https://zbmath.org/authors/?q=ai:diakonikolas.jelena"Fazel, Maryam"https://zbmath.org/authors/?q=ai:fazel.maryam"Orecchia, Lorenzo"https://zbmath.org/authors/?q=ai:orecchia.lorenzoThe improved distributed algorithms are obtained for constructing \(\varepsilon\)-approximate solutions to \(\alpha\)-fair packing problems. For \(\alpha\in [0,1)\), one of the main results (Theorem 4.4) shows that a solution with a \((1+\varepsilon)\)-relative error is reached within \(O(\log(n\rho)\log(mn\rho/\varepsilon)/(1-\alpha)^3\varepsilon^2)\) iterations. For \(\alpha=1\), one of the main results (Theorem 4.8) yields a \(\varepsilon\)-approximate convergence in \(O(\log^3(mn\rho/\varepsilon)/\varepsilon^2)\) iterations. For \(\alpha>1\), one of the main results (Theorem 4.14) shows that a solution with a \((1-\varepsilon)\)-relative error is reached within
\(O(\max\{\alpha^3\log(n\rho/\varepsilon)\log(mn\rho/\varepsilon)/\varepsilon,\log(1/\varepsilon(\alpha-1))\log(mn\rho/\varepsilon)/\varepsilon(\alpha-1)\})\) iterations.
Reviewer: Yisheng Song (Hong Kong)An accelerated minimal gradient method with momentum for strictly convex quadratic optimizationhttps://zbmath.org/1491.901132022-09-13T20:28:31.338867Z"Oviedo, Harry"https://zbmath.org/authors/?q=ai:oviedo.harry"Dalmau, Oscar"https://zbmath.org/authors/?q=ai:dalmau.oscar"Herrera, Rafael"https://zbmath.org/authors/?q=ai:herrera.rafaelSummary: In this article we address the problem of minimizing a strictly convex quadratic function using a novel iterative method. The new algorithm is based on the well-known Nesterov's accelerated gradient method. At each iteration of our scheme, the new point is computed by performing a line-search scheme using a search direction given by a linear combination of three terms, whose parameters are chosen so that the residual norm is minimized at each step of the process. We establish the linear convergence of the proposed method and show that its convergence rate factor is analogous to the one available for other gradient methods. Finally, we present preliminary numerical results on some sets of synthetic and real strictly convex quadratic problems, showing that the proposed method outperforms in terms of efficiency, a wide collection of state-of-the art gradient methods, and that it is competitive against the conjugate gradient method in terms of CPU time and number of iterations.Local convergence of tensor methodshttps://zbmath.org/1491.901172022-09-13T20:28:31.338867Z"Doikov, Nikita"https://zbmath.org/authors/?q=ai:doikov.nikita"Nesterov, Yurii"https://zbmath.org/authors/?q=ai:nesterov.yuriiSummary: In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.Accelerated methods with fastly vanishing subgradients for structured non-smooth minimizationhttps://zbmath.org/1491.901192022-09-13T20:28:31.338867Z"Maingé, Paul-Emile"https://zbmath.org/authors/?q=ai:mainge.paul-emile"Labarre, Florian"https://zbmath.org/authors/?q=ai:labarre.florianSummary: In a real Hilbert space, we study a new class of forward-backward algorithms for structured non-smooth minimization problems. As a special case of the parameters, we recover the method AFB (\textit{Accelerated Forward-Backward}) that was recently discussed as an enhanced variant of FISTA (\textit{Fast Iterative Soft Thresholding Algorithm}). Our algorithms enjoy the well-known properties of AFB. Namely, they generate convergent sequences \((x_n)\) that minimize the function values at the rate \(o(n^{-2})\). Another important specificity of our processes is that they can be regarded as discrete models suggested by first-order formulations of Newton-like dynamical systems. This permit us to extend to the non-smooth setting, a property of fast convergence to zero of the gradients, established so far for discrete Newton-like dynamics with smooth potentials only. In specific, as a new result, we show that the latter property also applies to AFB. To prove this stability phenomenon, we develop a technical analysis that can be also useful regarding many other related developments. Numerical experiments are furthermore performed so as to illustrate the properties of the considered algorithms comparing with other existing ones.An active-set algorithm for norm constrained quadratic problemshttps://zbmath.org/1491.901322022-09-13T20:28:31.338867Z"Rontsis, Nikitas"https://zbmath.org/authors/?q=ai:rontsis.nikitas"Goulart, Paul J."https://zbmath.org/authors/?q=ai:goulart.paul-j"Nakatsukasa, Yuji"https://zbmath.org/authors/?q=ai:nakatsukasa.yujiSummary: We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by solving a series of trust-region subproblems (TRS). Underpinning the efficiency of this approach is that the global solution of the TRS has been widely studied in the literature, resulting in remarkably efficient algorithms and software. We extend these results by proving that nonglobal minimizers of the TRS, or a certificate of their absence, can also be calculated efficiently by computing the two rightmost eigenpairs of an eigenproblem. We demonstrate the usefulness and scalability of the algorithm in a series of experiments that often outperform state-of-the-art approaches; these include calculation of high-quality search directions arising in Sequential Quadratic Programming on problems of the \texttt{CUTEst} collection, and Sparse Principal Component Analysis on a large text corpus problem (70 million nonzeros) that can help organize documents in a user interpretable way.A subspace acceleration method for minimization involving a group sparsity-inducing regularizerhttps://zbmath.org/1491.901572022-09-13T20:28:31.338867Z"Curtis, Frank E."https://zbmath.org/authors/?q=ai:curtis.frank-e"Dai, Yutong"https://zbmath.org/authors/?q=ai:dai.yutong"Robinson, Daniel P."https://zbmath.org/authors/?q=ai:robinson.daniel-pA primal-dual algorithm for nonnegative \(N\)-th order CP tensor decomposition: application to fluorescence spectroscopy data analysishttps://zbmath.org/1491.901832022-09-13T20:28:31.338867Z"EL Qate, Karima"https://zbmath.org/authors/?q=ai:el-qate.karima"El Rhabi, Mohammed"https://zbmath.org/authors/?q=ai:el-rhabi.mohammed"Hakim, Abdelilah"https://zbmath.org/authors/?q=ai:hakim.abdelilah"Moreau, Eric"https://zbmath.org/authors/?q=ai:moreau.eric"Thirion-Moreau, Nadège"https://zbmath.org/authors/?q=ai:thirion-moreau.nadegeSummary: This work concerns the resolution of inverse problems encountered in multidimensional signal processing problems. Here, we address the problem of tensor decomposition, more precisely the Canonical Polyadic (CP) Decomposition also known as Parafac. Yet, when the amount of data is very large, this inverse problem may become numerically ill-posed and consequently hard to solve. This requires to introduce a priori information about the system, often in the form of penalty functions. The difficulty that might arise is that the considered cost function may be non differentiable. It is the reason why we develop here a Primal-Dual Projected Gradient (PDPG) optimization algorithm to solve variational problems involving such cost functions. Moreover, despite its theoretical importance, most approaches of the literature developed for nonnegative CP decomposition, have not actually addressed the convergence issue. Furthermore, most methods with convergence rate guarantee require restrictive conditions for their theoretical analyses. Hence, a theoretical void still exists for the convergence rate problem under very mild conditions. Therefore, the two main aims of this work are (i) to design a CP-PDPG algorithm and then (ii) to prove, under mild conditions, its convergence to the set of minimizers at the rate \(O(1/k)\), \(k\) being the iteration number. The effectiveness and the robustness of the proposed approach are illustrated through numerical examples.Zeroth-order regularized optimization (ZORO): approximately sparse gradients and adaptive samplinghttps://zbmath.org/1491.901842022-09-13T20:28:31.338867Z"Cai, HanQin"https://zbmath.org/authors/?q=ai:cai.hanqin"McKenzie, Daniel"https://zbmath.org/authors/?q=ai:mckenzie.daniel"Yin, Wotao"https://zbmath.org/authors/?q=ai:yin.wotao"Zhang, Zhenliang"https://zbmath.org/authors/?q=ai:zhang.zhenliangA probabilistic numerical method for a class of mean field gameshttps://zbmath.org/1491.910222022-09-13T20:28:31.338867Z"Sahar, Ben Aziza"https://zbmath.org/authors/?q=ai:sahar.ben-aziza"Salwa, Toumi"https://zbmath.org/authors/?q=ai:salwa.toumiSpatial and color hallucinations in a mathematical model of primary visual cortexhttps://zbmath.org/1491.920102022-09-13T20:28:31.338867Z"Faugeras, Olivier D."https://zbmath.org/authors/?q=ai:faugeras.olivier-d"Song, Anna"https://zbmath.org/authors/?q=ai:song.anna"Veltz, Romain"https://zbmath.org/authors/?q=ai:veltz.romainThere is a study on chromatic aspects of visual perception by means of neural fields theory. The article considers a neural field model for color perceptions introduced by the authors in a previous article, [\textit{A. Song} et al., ``A neural field model for color perception unifying assimilation and contrast'', PLoS Comput. Biol. 15, No. 6, Article ID e1007050, 28 p. (2019; \url{doi:10.1371/journal.pcbi.1007050})]. The investigation here focuses on how this model can predict visual hallucinations. The model is briefly presented in the second section and is described by an initial value problem to the Hammerstein equation which is an integro-partial differential equation for the average membrane potential V (r, c, t). Under some assumptions and choice of an appropriate function space, one reminds that the existence of a unique solution to the Cauchy problem has been proved in an earlier paper from, [\textit{R. Veltz} and \textit{O. Faugeras}, SIAM J. Appl. Dyn. Syst. 9, No. 3, 954--998 (2010; Zbl 1194.92015)]. The aim further is to study the stationary solutions to the considered equations and their bifurcations which are interpreted as possible metaphors of visual hallucinations. In the next three sections one introduces the notion of stationary solutions, their bifurcations, one performs the computation of the spectrum of the linear operator in the neural model and one shows the symmetries of the model and equivariant bifurcations of the solutions. Examples of four types of planforms are presented in the sixth Section. A numerical bifurcation analysis is performed in the seventh Section. One describes the numerical experiments and presents some first, second and third bifurcation diagrams. Conclusions are provided in the eighth Section.
A very interesting research paper and good presentation.
Reviewer: Claudia Simionescu-Badea (Wien)Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equationhttps://zbmath.org/1491.920772022-09-13T20:28:31.338867Z"Tijani, Yusuf O."https://zbmath.org/authors/?q=ai:tijani.yusuf-o"Appadu, Appanah R."https://zbmath.org/authors/?q=ai:appadu.appanah-raoSummary: The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher's equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodeshttps://zbmath.org/1491.920802022-09-13T20:28:31.338867Z"Harrach, Bastian"https://zbmath.org/authors/?q=ai:harrach.bastianModeling the COVID-19 pandemic: a primer and overview of mathematical epidemiologyhttps://zbmath.org/1491.921212022-09-13T20:28:31.338867Z"Saldaña, Fernando"https://zbmath.org/authors/?q=ai:saldana.fernando"Velasco-Hernández, Jorge X."https://zbmath.org/authors/?q=ai:velasco-hernandez.jorge-xSummary: Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, the impact of non-pharmaceutical interventions in the control of the pandemic, vaccine deployment, herd-immunity, viral evolution and the possibility of vaccine escape. From the perspective of mathematical epidemiology, we highlight the important properties, findings, and, of course, deficiencies, that all these models have.Discrete dynamical models on \textit{Wolbachia} infection frequency in mosquito populations with biased release ratioshttps://zbmath.org/1491.921242022-09-13T20:28:31.338867Z"Shi, Yantao"https://zbmath.org/authors/?q=ai:shi.yantao"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo|zheng.bo.1Summary: We develop two discrete models to study how supplemental releases affect the \textit{Wolbachia} spreading dynamics in cage mosquito populations. The first model focuses on the case when only infected males are released at each generation. This release strategy has been proved to be capable of speeding up the \textit{Wolbachia} persistence by suppressing the compatible matings between uninfected individuals. The second model targets the case when only infected females are released at each generation. For both models, detailed model formulation, enumeration of the positive equilibria and their stability analysis are provided. Theoretical results show that the two models can generate bistable dynamics when there are three positive equilibrium points, semi-stable dynamics for the case of two positive equilibrium points. And when the positive equilibrium point is unique, it is globally asymptotically stable. Some numerical simulations are offered to get helpful implications on the design of the release strategy.Anisotropic smoothness classes: from finite element approximation to image modelshttps://zbmath.org/1491.940082022-09-13T20:28:31.338867Z"Mirebeau, Jean-Marie"https://zbmath.org/authors/?q=ai:mirebeau.jean-marie"Cohen, Albert"https://zbmath.org/authors/?q=ai:cohen.albertSummary: We propose and study quantitative measures of smoothness \(f \rightarrowtail A(f)\) which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest example being \(A_{p}(f)=\|\sqrt{|\text{det}(d^{2}f)|}\|_{L^{\tau}}\) which appears when approximating in the \(L p\) norm by piecewise linear elements when \(\frac{1}{\tau}=\frac{1}{p}+1\). The quantities \(A(f)\) are not semi-norms, and therefore cannot be used to define linear function spaces. We show that these quantities can be well defined by mollification when \(f\) has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used total variation semi-norm which does not account for the smoothness of the edges.Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parametershttps://zbmath.org/1491.940092022-09-13T20:28:31.338867Z"Na, Hanwool"https://zbmath.org/authors/?q=ai:na.hanwool"Kang, Myeongmin"https://zbmath.org/authors/?q=ai:kang.myeongmin"Jung, Miyoun"https://zbmath.org/authors/?q=ai:jung.miyoun"Kang, Myungjoo"https://zbmath.org/authors/?q=ai:kang.myungjooSummary: In this article, we introduce a novel variational model for the restoration of images corrupted by multiplicative Gamma noise. The model incorporates a convex data-fidelity term with a nonconvex version of the total generalized variation (TGV). In addition, we adopt a spatially adaptive regularization parameter (SARP) approach. The nonconvex TGV regularization enables the efficient denoising of smooth regions, without staircasing artifacts that appear on total variation regularization-based models, and edges and details to be conserved. Moreover, the SARP approach further helps preserve fine structures and textures. To deal with the nonconvex regularization, we utilize an iteratively reweighted \(\ell_1\) algorithm, and the alternating direction method of multipliers is employed to solve a convex subproblem. This leads to a fast and efficient iterative algorithm for solving the proposed model. Numerical experiments show that the proposed model produces better denoising results than the state-of-the-art models.Multi-modality image registration models and efficient algorithmshttps://zbmath.org/1491.940142022-09-13T20:28:31.338867Z"Zhang, Daoping"https://zbmath.org/authors/?q=ai:zhang.daoping"Theljani, Anis"https://zbmath.org/authors/?q=ai:theljani.anis"Chen, Ke"https://zbmath.org/authors/?q=ai:chen.keSummary: In this Chapter we discuss multi-modality image registration models and efficient algorithms. We propose a simple method to enhance a variational model to generate a diffeomorphic transformation. The idea is illustrated by using a particular model based on reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. By adding a control term motivated by quasi-conformal maps and Beltrami coefficients, the model has the ability to guarantee a diffeomorphic transformation. Without this feature, the model may lead to visually pleasing but invalid results. To solve the model numerically, we present both a Gauss-Newton method and an augmented Lagrangian method to solve the resulting discrete optimization problem. A multilevel technique is employed to speed up the initialization and reduce the possibility of getting local minima of the underlying functional. Finally numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.
For the entire collection see [Zbl 1476.68015].Remainder and quotient without polynomial long divisionhttps://zbmath.org/1491.970052022-09-13T20:28:31.338867Z"Laudano, Francesco"https://zbmath.org/authors/?q=ai:laudano.francescoSummary: We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in \(\mathbb Z[x]\) any integral polynomial with degree less than 6. The arguments are suitable for building classroom/homework activities in basic algebra courses.A problem-based learning proposal to teach numerical and analytical nonlinear root searching methodshttps://zbmath.org/1491.970442022-09-13T20:28:31.338867Z"González-Santander, Juan Luis"https://zbmath.org/authors/?q=ai:gonzalez-santander.juan-luis"Sánchez-Lasheras, Fernando"https://zbmath.org/authors/?q=ai:sanchez-lasheras.fernando(no abstract)