Recent zbMATH articles in MSC 65Dhttps://zbmath.org/atom/cc/65D2024-11-01T15:51:55.949586ZUnknown authorWerkzeugComputer aided algebraic geometry: constructing surfaces of genus zerohttps://zbmath.org/1544.140592024-11-01T15:51:55.949586Z"Pignatelli, Roberto"https://zbmath.org/authors/?q=ai:pignatelli.robertoSummary: Everybody knows that mathematics has a key role in the development of the modern technology. It is less known that the modern technology gives something back to mathematics. In this note we give an account on how the combination of classical results as the Riemann Existence Theorem with the use of computers and computational algebra programs answered interesting old-standing problems in classical algebraic geometry, namely regarding the construction and the classification of new surfaces of general type. We also give a full list of the surfaces constructed with this method up to now, and present the next challenges on the subject.
For the entire collection see [Zbl 1293.00025].Discrete-velocity-direction models of BGK-type with minimum entropy. II: Weighted modelshttps://zbmath.org/1544.351732024-11-01T15:51:55.949586Z"Chen, Yihong"https://zbmath.org/authors/?q=ai:chen.yihong"Huang, Qian"https://zbmath.org/authors/?q=ai:huang.qian"Yong, Wen-An"https://zbmath.org/authors/?q=ai:yong.wen-anThe Bathnagar-Gross-Krook (BGK) model is an approximation of the Boltzmann equation, describing the time evolution of a single monoatomic rarefied gas and satisfying the properties: conservation and entropy inequality. The BGK equation for the density function \(f=f(t,x,\xi ,\zeta )\) has the form \(\partial_tf+\xi\cdot \nabla_xf=\tau^{-1} (\mathcal{E}[f]-f)\), where \((x,\xi )\in \mathbb{R}^D\times \mathbb{R}^D\) with \(D=2\) or 3, \(\zeta\in \mathbb{R}^L\) represents the possible internal molecular degrees of freedom, and \(\tau \) is a characteristic collision time. Here, \(\mathcal{E}[f]\) is the local equilibrium state. In the present paper the authors consider a discrete-velocity-direction model (DVDM) with collisions of BGK-type for simulating gas flows, where the molecular motion is confined to some prescribed directions but the speed is still a continuous variable in each direction. The BGK-DVDM is improved in two aspects. First, the internal molecular degrees of freedom is included so that more realistic fluid properties can be realized. The authors introduce a weighted function in each orientation when recovering the macroscopic parameters, as opposed to the previous treatment. For the new weighted DVDM, the established properties of the well-behaved discrete equilibrium still hold. The DVDM is considered under requirement that the molecule transport is limited to \(N\) prescribed directions \(\{\boldsymbol{l}_m\}_{m=1}^{N}\) with each \(\boldsymbol{l}_m\) located on the unit sphere \(\mathbb{S}^{D-1}\), but the velocity magnitude \(\xi\in \mathbb{R}\) in each direction remains continuous. There is a pair of requirements: 1) \((\boldsymbol{l}_1,\dots, \boldsymbol{l}_N)\in \mathbb{R}^{D\times N}\) is of rank \(D\) and therefore \(N\geq D\); 2) Each direction \(\boldsymbol{l}_m\) and its opposite \(-\boldsymbol{l}_m\) belong to \(S_m\subset\mathbb{S}^{D-1}\), where the quantities \(S_m\) constitute a disjoint partition of the unit sphere \(\mathbb{S}^{D-1}= \bigcup_{m=1}^{N}S_m\). Each \(S_m\) has the same measure. Note that the distribution \(f\) is replaced by \(N\) distributions \(\{f_m(t,x,\xi ,\zeta )\}_{m=1}^{N}\) with \(\xi\in\mathbb{R}\) and \(\zeta\in\mathbb{R}^L\). The transport velocity for \(f_m\) is \(\xi\boldsymbol{l}_m\), and the governing equation for \(f_m\) becomes \(\partial_tf_m+\xi\boldsymbol{l}_m\cdot \nabla_xf_m = \tau^{-1} (\mathcal{E}_m-f_m)\) \((m=1,\dots,N)\) with the local equilibriums \(\mathcal{E}_m\). The authors introduce a weighted function in each orientation when recovering the macroscopic parameters. With the weighted DVDM, the authors consider three submodels by incorporating the discrete velocity method, the Gaussian-extended quadrature method of moments and the Hermite spectral method in each direction. It seems the stated spatial-time submodels are multidimensional versions corresponding to the three approaches. Some numerical tests with a series of 1-D and 2-D flow problems show the efficiency of the weighted DVDM.
For Part I see [ibid. 95, No. 3, Paper No. 80, 29 p. (2023; Zbl 1515.65235)].
Reviewer: Dimitar A. Kolev (Sofia)Spectral decomposition of \(H^1 (\mu)\) and Poincaré inequality on a compact interval -- application to kernel quadraturehttps://zbmath.org/1544.410192024-11-01T15:51:55.949586Z"Roustant, Olivier"https://zbmath.org/authors/?q=ai:roustant.olivier"Lüthen, Nora"https://zbmath.org/authors/?q=ai:luthen.nora"Gamboa, Fabrice"https://zbmath.org/authors/?q=ai:gamboa.fabriceLet \(\mathcal{B}\) be the set of probability distributions \(\mu\) whose density \(\rho\) is a positive \(C^1\) function on a finite real interval \([a,b]\). These are bounded perturbations of the uniform distribution \(\nu\) on \([a,b]\). The authors consider the integral \(I_{\mu}[f]:=\int_a^b f(x)d\mu(x)\) and the quadrature formulas (QF) \(I_{\mu_n}[f]:=\sum_{i=1}^n w_i f(x_i)\) where \(\mu_n\) is the discrete distribution with nodes \(X=(x_1,\ldots,x_n)\), \(x_i\in[a,b]\) and (positive) weights \(\mathbf{w}=(w_1,\ldots,w_n)\). In particular, they want to find the optimal QF minimising the worst case error \(\mathrm{wce}(\mu,\mu_n,\mathcal{H})= \sup_{f\in\mathcal{H}, \|f\|_{\mathcal{H}}\le1}|(I_\mu-I_{\mu_n})[f]|\). The integrand \(f\) belongs to some space \(\mathcal{H}\) which can be a Sobolev space \(H^1(\mu)=\{f: f,f'\in L^2(\mu)\}\) which is shown to be a reproducing kernel Hilbert space (RKHS), and the QF are exact on finite dimensional subspaces of \(\mathcal{H}\).
Two approaches to construct the QF are considered and the links between both are proved, implying their equivalence:
\begin{itemize}
\item[(1)] A Chebyshev system (or T-system) \(\{u_k\}_{k\in\mathbb{N}}\) of eigenfunctions of the Sturm-Liouville operator \(L(f)=-(f'\rho)'+f\rho\) with boundary conditions \(f(a)=f(b)=0\) has properties similar to monomials \(x^k\) so that there is a unique \(n\)-point Gaussian-type formula with positive weights, exact in \(\mathrm{span}\{u_k\}_{k=0}^{2n-1}\). This is the (optimal) Poincaré QF in this subspace.
\item[(2)] Given the reproducing kernel \(K\), and the nodes \(X\), a basis \(K(x_i,\cdot)\) can be used and there exists an expression for the wce, so that the corresponding optimal \(\mathbf{w}\) can be obtained. The optimal kernel QF is the one that minimizes over all \(X\).
\end{itemize}
The expression for the error shows that convergence is proportional to \(n^{-1}\) and that the Poincaré QF is asymptotically optimal in \(\mathcal{H}\). More explicit formulas are obtained when \(\mu\) is the uniform distribution \(\nu\).
Numerical experiments suggest that the weighted Poincaré QF asymptotically behaves like for the uniform distribution with uniformly spaced nodes and \(\mathrm{wce}\sim \frac{1}{\sqrt{12}} n^{-1}\).
Reviewer: Adhemar Bultheel (Leuven)Exact and approximate solutions for mathematical models in science and engineering. Selected papers based on the presentations at the international conference on computational and mathematical methods in science and engineering, CMMSE, Costa Ballena, Rota, Cadiz, Spain, July 3--7, 2023https://zbmath.org/1544.650132024-11-01T15:51:55.949586ZPublisher's description: This contributed volume collects papers presented during a special session on integral methods in science and engineering at the 2023 International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE), held in Cadiz, Spain from July 3--8, 2023. Covering the applications of integral methods to scientific developments in a variety of fields, the chapters in this volume are written by well-known researchers in their respective disciplines and present new results in both pure and applied mathematics. Each chapter shares a common methodology based on a combination of analytic and computational tools, an approach that makes this collection a valuable, multidisciplinary reference on how mathematics can be applied to various real-world processes and phenomena.
The articles of this volume will be reviewed individually.Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemeshttps://zbmath.org/1544.650312024-11-01T15:51:55.949586Z"Gemignani, Luca"https://zbmath.org/authors/?q=ai:gemignani.luca"Romani, Lucia"https://zbmath.org/authors/?q=ai:romani.lucia"Viscardi, Alberto"https://zbmath.org/authors/?q=ai:viscardi.albertoSummary: The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes.Robust computation methods for sparse interpolation of multivariate polynomialshttps://zbmath.org/1544.650322024-11-01T15:51:55.949586Z"Kondo, Kazuki"https://zbmath.org/authors/?q=ai:kondo.kazuki"Sekigawa, Hiroshi"https://zbmath.org/authors/?q=ai:sekigawa.hiroshiRobust algorithms for sparse interpolation of multivariate polynomialshttps://zbmath.org/1544.650332024-11-01T15:51:55.949586Z"Numahata, Dai"https://zbmath.org/authors/?q=ai:numahata.dai"Sekigawa, Hiroshi"https://zbmath.org/authors/?q=ai:sekigawa.hiroshiA generalized spatially adaptive sparse grid combination technique with dimension-wise refinementhttps://zbmath.org/1544.650342024-11-01T15:51:55.949586Z"Obersteiner, Michael"https://zbmath.org/authors/?q=ai:obersteiner.michael"Bungartz, Hans-Joachim"https://zbmath.org/authors/?q=ai:bungartz.hans-joachimSummary: Today, high-dimensional calculations can be found in almost all scientific disciplines. The application of machine learning and uncertainty quantification methods are common examples where high-dimensional problems appear. Typically, these problems are computationally expensive or even infeasible on current machines due to the curse of dimensionality. The sparse grid combination technique is one method to mitigate this effect, but it still does not generate optimal grids for many application scenarios. In such cases, adaptivity strategies are applied to further optimize the grid generation. Generally, adaptive grid generation strategies can be be classified as spatially or dimensionally adaptive. One of the most prominent examples is the dimension-adaptive combination technique, which is easy to implement and suitable for cases where dimensions contribute in different magnitudes to the accuracy of the solution. Unfortunately, spatial adaptivity is not possible in the standard combination technique due to the required regular structure of the grids. We therefore propose a new algorithmic variant of the combination technique that is based on the combination of rectilinear grids which can adapt themselves locally to the target function using 1-dimensional refinements. We further increase the efficiency by adjusting point levels with tree rebalancing. Results for numerical quadrature and interpolation show that we can significantly improve upon the standard combination technique and compete with or even surpass common spatially adaptive implementations with sparse grids. At the same time our method keeps the black-box property of the combination technique which makes it possible to apply it to black-box solvers.Solving the problem of interpreting observations using the spline approximation of the scanned functionhttps://zbmath.org/1544.650352024-11-01T15:51:55.949586Z"Verlan, A. F."https://zbmath.org/authors/?q=ai:verlan.a-f"Malachivskyy, P. S."https://zbmath.org/authors/?q=ai:malachivskyy.petro-s"Pizyur, Ya. V."https://zbmath.org/authors/?q=ai:pizyur.ya-vSummary: An accuracy analysis of the numerical implementation of the frequency method for solving the integral equation in the problem of interpreting technical observations using the spline approximation of the scanned function is presented. The algorithm for solving the integral equation of the interpretation problem, which is based on the application of the Tikhonov regularization method with the search for a solution in the frequency domain with a truncation of the frequency spectrum is investigated. To increase the accuracy of the interpretation results, the use of spline approximation of the values of the scanned function, i.e., the right-hand side of the integral equation, is proposed. The accuracy of the solution of the integral equation is estimated using the regularization method and taking into account the error accompanied by the inaccuracy of the right-hand side, as well as the error in calculating the kernel values. A method for calculating the accuracy-optimal smoothing spline approximation of the scanned function is proposed.Highly localized RBF Lagrange functions for finite difference methods on sphereshttps://zbmath.org/1544.650362024-11-01T15:51:55.949586Z"Erb, W."https://zbmath.org/authors/?q=ai:erb.wolfgang"Hangelbroek, T."https://zbmath.org/authors/?q=ai:hangelbroek.thomas-c"Narcowich, F. J."https://zbmath.org/authors/?q=ai:narcowich.francis-j"Rieger, C."https://zbmath.org/authors/?q=ai:rieger.christian"Ward, J. D."https://zbmath.org/authors/?q=ai:ward.joseph-dThe authors study how rapidly decaying RBF Lagrange functions on the sphere can be used to create a numerically feasible, stable finite difference method based on radial basis functions (an RBF-FD-like method) and the application to certain PDE. The RBF-FD method and its variants are modifications of the classical finite difference method suitable for working with unstructured point sets. The authors consider a time-independent partial differential equation on a manifold without boundary (such as the sphere \(\mathbb{S}^2\)), \(\mathscr Lu = f\). The PDE is replaced by a linear system \(\mathbf Mu = y\) which can then be solved for a discrete solution. The aim of this paper is to give conditions on the operator \(\mathscr L\) which guarantee stable invertibility of \(\mathbf M\), and provide satisfying convergence rates.
Reviewer: Antonio López-Carmona (Granada)A new algorithm for calculating Padé approximants and its Matlab implementationhttps://zbmath.org/1544.650372024-11-01T15:51:55.949586Z"Ibryaeva, Ol'ga Leonidovna"https://zbmath.org/authors/?q=ai:ibryaeva.olga-leonidovnaSummary: A new algorithm for calculating a Pade approximant is proposed. The algorithm is based on the choice of the Pade approximant's denominator of least degree. It is shown that the new algorithm does not lead to the appearance of the Froissart doublets in contrast to available procedures for calculating Pade approximants in Maple and Mathematica.Parametrization of the bisector of two low degree surfaceshttps://zbmath.org/1544.650382024-11-01T15:51:55.949586Z"Adamou, Ibrahim"https://zbmath.org/authors/?q=ai:adamou.ibrahim"Fioravanti, Mario"https://zbmath.org/authors/?q=ai:fioravanti.mario-a"Gonzalez-Vega, Laureano"https://zbmath.org/authors/?q=ai:gonzalez-vega.laureanoSummary: The bisectors are geometric constructions with different applications in tool path generation, motion planning, NC-milling, etc. We present a new approach to determine an algebraic representation (parameterization or implicit equation) of the bisector surface of two given low degree parametric surfaces. The method uses the so-called generalized Cramer rules, and suitable elimination steps. The new introduced approach allows to easily obtain parameterizations of the plane-quadric, plane-torus, circular cylinder-quadric, circular cylinder-torus, cylinder-cylinder, cylinder-cone and cone-cone bisectors, which are rational in most cases. In the remaining cases the parametrization involves one square root, which is well suited for a good approximation of the bisector.
For the entire collection see [Zbl 1293.00025].Font design through RQT Bézier curve using shape parametershttps://zbmath.org/1544.650392024-11-01T15:51:55.949586Z"Dube, Mridula"https://zbmath.org/authors/?q=ai:dube.mridula"Gupta, Nishi"https://zbmath.org/authors/?q=ai:gupta.nishiSummary: To effectively draw the free form curves, we studied RQT (rational quintic trigonometric) Bézier curve using two shape parameters. These new curves are more adjustable because of the existence of shape parameters and geometric characteristics. To confirm whether the proposed curve fulfilled the convex hull property or not, we used limitations on shape and weight, and declared end-pointed curvatures. Local adjustments to the curve shape can be made by modifying its weights and shape parameter values. This curve is utilized for smooth curve compositions by generating piecewise rational trigonometric curves that are adjacent to parametric and geometric Hermite continuity criteria. Then, as an example in font design, piecewise curves are utilized to draw the structure of Devanagari alphabets.
For the entire collection see [Zbl 1522.00198].Performance analysis and optimisation of spatially-varying infill microstructure within CAD geometrieshttps://zbmath.org/1544.650402024-11-01T15:51:55.949586Z"Ma, Chuang"https://zbmath.org/authors/?q=ai:ma.chuang"Zhang, Jianhao"https://zbmath.org/authors/?q=ai:zhang.jianhao"Zhu, Yichao"https://zbmath.org/authors/?q=ai:zhu.yichaoSummary: The article is aimed to provide an effective digital solution in support of the infill microstructural design within objectives generated from normal computer aided design (CAD) systems. The key here is to deploy the microstructure and to analyse the performance of the resulting multiscale objective in the parameter domain, from which a CAD objective is mapped, usually by means of Non-Uniform Rational B-Splines (NURBS) functions. In the regularly shaped domain defined in the parameter space, another set of mapping functions are defined for the description of spatially-varying microstructure (in the parameter space), and the actual multiscale CAD objective gets represented through composition of these mapping functions with the NURBS-based functions for CAD geometry generation. When the introduced mapping functions are also NURBS-based, the proposed representation strategy becomes more favoured by CAD systems, in the sense that the CAD geometry and its infill can be tuned simultaneously but independently with a same set of NURBS basis. Moreover, our computer-aided engineering (CAE) module is also installed over the parameter domain, based on a machine-learning-based homogenisation formulation for compliance and strength analysis. Thus the finite element meshes stay unchanged although the accommodating CAD geometry keeps varying. With the aforementioned treatments, a consecutive CAD-consistent scheme is proposed. Our numerical results show that each time as the control points are moved (by a CAD designer), one may choose to wait several seconds for two-dimensional cases and/or a few minutes for three-dimensional cases, to get a CAD geometry filled with microstructure bearing optimised compliance.A novel generalization of trigonometric Bézier curve and surface with shape parameters and its applicationshttps://zbmath.org/1544.650412024-11-01T15:51:55.949586Z"Maqsood, Sidra"https://zbmath.org/authors/?q=ai:maqsood.sidra"Abbas, Muhammad"https://zbmath.org/authors/?q=ai:abbas.muhammad-mohsin"Hu, Gang"https://zbmath.org/authors/?q=ai:hu.gang.2"Ramli, Ahmad Lutfi Amri"https://zbmath.org/authors/?q=ai:ramli.ahmad-lutfi-amri"Miura, Kenjiro T."https://zbmath.org/authors/?q=ai:miura.kenjiro-takaiSummary: Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. These basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. The generalized trigonometric Bézier (or GT-Bézier, for short) curves and surfaces are defined on these basis functions and also analyze their geometric properties which are analogous to classical Bézier curves and surfaces. This analysis shows that the existence of shape parameters brings a convenience to adjust the shape of the curve and surface by simply modifying their values. These GT-Bézier curves meet the conditions required for parametric continuity (\( C^0\), \(C^1\), \(C^2\), and \(C^3\)) as well as for geometric continuity (\( G^0\), \(G^1\), and \(G^2\)). Furthermore, some curve and surface design applications have been discussed. The demonstrating examples clarify that the new curves and surfaces provide a flexible approach and mathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.Topology analysis of global and local RBF transformations for image registrationhttps://zbmath.org/1544.650422024-11-01T15:51:55.949586Z"Cavoretto, Roberto"https://zbmath.org/authors/?q=ai:cavoretto.roberto"De Rossi, Alessandra"https://zbmath.org/authors/?q=ai:de-rossi.alessandra"Qiao, Hanli"https://zbmath.org/authors/?q=ai:qiao.hanliSummary: For elastic registration, topology preservation is a necessary condition to be satisfied, especially for landmark-based image registration. In this paper, we focus on the topology preservation properties of two different families of radial basis functions (RBFs), known as Gneiting and Matérn functions. Firstly, we consider a small number of landmarks, dealing with the cases of one, two and four landmark matching; in all these situations we analyze topology preservation and compare numerical results with those obtained by Wendland functions. Secondly, we discuss the registration properties of these two families of functions, when we have a larger number of landmarks. Finally, we analyze the behavior of Gneiting and Matérn functions, considering some test examples known in the literature and a real application.Nonnegative moment coordinates on finite element geometrieshttps://zbmath.org/1544.650432024-11-01T15:51:55.949586Z"Dieci, L."https://zbmath.org/authors/?q=ai:dieci.luca"Difonzo, Fabio V."https://zbmath.org/authors/?q=ai:difonzo.fabio-v"Sukumar, N."https://zbmath.org/authors/?q=ai:sukumar.natarajanSummary: In this paper, we introduce new generalized barycentric coordinates (coined as \textit{moment coordinates}) on convex and nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with \(n\) vertices (nodes) in \(\mathbb{R}^2\), the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank \(n\), whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties.Adaptive image compression via optimal mesh refinementhttps://zbmath.org/1544.650442024-11-01T15:51:55.949586Z"Feischl, Michael"https://zbmath.org/authors/?q=ai:feischl.michael"Hackl, Hubert"https://zbmath.org/authors/?q=ai:hackl.hubertSummary: The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient compression algorithm. We demonstrate experimentally, that the new algorithm can achieve better compression ratios than standard JPEG compression with no visible loss of quality on many images. The mathematical core of this work shows that Binev's optimal tree approximation algorithm is applicable to image compression with high probability, when we assume small additive Gaussian noise on the pixels of the image.Extrapolation algorithm for computing multiple Cauchy principal value integralshttps://zbmath.org/1544.650452024-11-01T15:51:55.949586Z"Li, Jin"https://zbmath.org/authors/?q=ai:li.jin.2"Cheng, Yongling"https://zbmath.org/authors/?q=ai:cheng.yonglingSummary: In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by the middle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.Corrigendum to: ``The Lagrange-mesh method''https://zbmath.org/1544.650462024-11-01T15:51:55.949586Z"Baye, Daniel"https://zbmath.org/authors/?q=ai:baye.danielCorrigendum to the author's paper [ibid. 565, 1--107 (2015; Zbl 1357.65031)].Asymptotic variance of Newton-Cotes quadratures based on randomized sampling pointshttps://zbmath.org/1544.650472024-11-01T15:51:55.949586Z"Stehr, Mads"https://zbmath.org/authors/?q=ai:stehr.mads"Kiderlen, Markus"https://zbmath.org/authors/?q=ai:kiderlen.markusSummary: We consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton-Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton-Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator.Quadrature-free polytopic discontinuous Galerkin methods for transport problemshttps://zbmath.org/1544.652102024-11-01T15:51:55.949586Z"Radley, Thomas J."https://zbmath.org/authors/?q=ai:radley.thomas-j"Houston, Paul"https://zbmath.org/authors/?q=ai:houston.paul"Hubbard, Matthew E."https://zbmath.org/authors/?q=ai:hubbard.matthew-eSummary: In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two and three dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its derivatives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to compute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.Simultaneous approximation of Hilbert and Hadamard transforms on bounded intervalshttps://zbmath.org/1544.652212024-11-01T15:51:55.949586Z"Mezzanotte, Domenico"https://zbmath.org/authors/?q=ai:mezzanotte.domenico"Occorsio, Donatella"https://zbmath.org/authors/?q=ai:occorsio.donatellaThis interesting paper under review studies the simultaneous approximation of the weighted Hilbert and Hadamard transforms of a fixed function \(f:(-1,1)\to \mathbb R\) with suitable smoothness and typically satisfying \(|(fw)(x)|\to 0,\, x\to \pm 1\) where the weight \(w\) is a Jacobi weight with certain parameters. Given such a fixed function \(f\) and weight \(w\), define the weighted Hilbert transform of \(f\) and weighted Hadamard transform of \(f\) respectively by
\[
H_0^w(f,x):=\lim_{\varepsilon\to 0}\int_{|t-x|>\varepsilon}\frac{f(t)w(t)}{t-x}dt,\, x\in (-1,1).
\]
\[
H_1^w(x):=\frac{d}{dx}H_0^w(f,x).
\]
These operators have numerous applications in areas diverse as integral equations, signal processing, data science, harmonic analysis and approximation theory. One classical approximation technique is to approximate well the function \(f\) by a Lagrange interpolation polynomial of certain degree where the interpolation is with respect to the zeroes of the orthonormal polynomials with respect to the weight \(w\) and then applying a scheme of product integration rules. The authors propose a scheme of product integration rules for the simultaneous approximation of \(H_0^w\) and \(H_1^w\) for different classes of functions \(f\) and Jacobi weights \(w\). The advantages of such a scheme are for example a saving in the number of function evaluations and the avoidance of the derivatives of the density function \(f\) when approximating the Hadamard transform. Stability and convergence results are obtained and theoretical estimates are confirmed by numerical tests.
The paper is well written.
Reviewer: Steven B. Damelin (Ann Arbor)Derivative-free separable quadratic modeling and cubic regularization for unconstrained optimizationhttps://zbmath.org/1544.901842024-11-01T15:51:55.949586Z"Custódio, A. L."https://zbmath.org/authors/?q=ai:custodio.ana-luisa"Garmanjani, R."https://zbmath.org/authors/?q=ai:garmanjani.rohollah"Raydan, M."https://zbmath.org/authors/?q=ai:raydan.marcosSummary: We present a derivative-free separable quadratic modeling and cubic regularization technique for solving smooth unconstrained minimization problems. The derivative-free approach is mainly concerned with building a quadratic model that could be generated by numerical interpolation or using a minimum Frobenius norm approach, when the number of points available does not allow to build a complete quadratic model. This model plays a key role to generate an approximated gradient vector and Hessian matrix of the objective function at every iteration. We add a specialized cubic regularization strategy to minimize the quadratic model at each iteration, that makes use of separability. We discuss convergence results, including worst case complexity, of the proposed schemes to first-order stationary points. Some preliminary numerical results are presented to illustrate the robustness of the specialized separable cubic algorithm.Isogonal embeddings of interwoven and self-entangled honeycomb (\(\mathbf{hcb}\)) nets and related interpenetrating primitive cubic (\(\mathbf{pcu}\)) netshttps://zbmath.org/1544.922392024-11-01T15:51:55.949586Z"O'Keeffe, Michael"https://zbmath.org/authors/?q=ai:okeeffe.michael"Treacy, Michael M. J."https://zbmath.org/authors/?q=ai:treacy.michael-m-jSummary: Two- and three-periodic vertex-transitive (isogonal) piecewise-linear embeddings of self-entangled and interwoven honeycomb nets are described. The infinite families with trigonal symmetry and edge transitivity (isotoxal) are particularly interesting as they have the Borromean property that no two nets are directly linked. These also lead directly to infinite families of interpenetrating primitive cubic nets (\(\mathbf{pcu}\)) that are also vertex- and edge-transitive and have embeddings with \(90^\circ\) angles between edges.
{\copyright} 2023 O'Keeffe and Treacy. \textit{Acta Crystallographica Section A. Foundations and Advances} published by IUCr Journals.Application of improved simplex quadrature cubature Kalman filter in nonlinear dynamic systemhttps://zbmath.org/1544.938062024-11-01T15:51:55.949586Z"Cao, Ting"https://zbmath.org/authors/?q=ai:cao.ting"Gao, Huo-tao"https://zbmath.org/authors/?q=ai:gao.huotao"Sun, Chun-feng"https://zbmath.org/authors/?q=ai:sun.chun-feng"Ling, Yun"https://zbmath.org/authors/?q=ai:ling.yun"Ru, Guo-bao"https://zbmath.org/authors/?q=ai:ru.guobaoSummary: A novel spherical simplex Gauss-Laguerre quadrature cubature Kalman filter is proposed to improve the estimation accuracy of nonlinear dynamic system. The nonlinear Gaussian weighted integral has been approximately evaluated using the spherical simplex rule and the arbitrary order Gauss-Laguerre quadrature rule. Thus, a spherical simplex Gauss-Laguerre cubature quadrature rule is developed, from which the general computing method of the simplex cubature quadrature points and the corresponding weights are obtained. Then, under the nonlinear Kalman filtering framework, the spherical simplex Gauss-Laguerre quadrature cubature Kalman filter is derived. A high-dimensional nonlinear state estimation problem and a target tracking problem are utilized to demonstrate the effectiveness of the proposed spherical simplex Gauss-Laguerre cubature quadrature rule to improve the performance.Hyperspectral images denoising via nonconvex regularized low-rank and sparse matrix decompositionhttps://zbmath.org/1544.941262024-11-01T15:51:55.949586Z"Xie, Ting"https://zbmath.org/authors/?q=ai:xie.ting"Li, Shutao"https://zbmath.org/authors/?q=ai:li.shutao"Sun, Bin"https://zbmath.org/authors/?q=ai:sun.binEditorial remark: No review copy delivered.Infrared small target detection with total variation and reweighted \(\ell_1\) regularizationhttps://zbmath.org/1544.941892024-11-01T15:51:55.949586Z"Fang, Houzhang"https://zbmath.org/authors/?q=ai:fang.houzhang"Chen, Min"https://zbmath.org/authors/?q=ai:chen.min.6|chen.min.3|chen.min.1|chen.min.5"Liu, Xiyang"https://zbmath.org/authors/?q=ai:liu.xiyang"Yao, Shoukui"https://zbmath.org/authors/?q=ai:yao.shoukuiSummary: Infrared small target detection plays an important role in infrared search and tracking systems applications. It is difficult to perform target detection when only a single image with complex background clutters and noise is available, where the key is to suppress the complex background clutters and noise while enhancing the small target. In this paper, we propose a novel model for separating the background from the small target based on nonlocal self-similarity for infrared patch-image. A total variation-based regularization term for the small target image is incorporated into the model to suppress the residual background clutters and noise while enhancing the smoothness of the solution. Furthermore, a reweighted sparse constraint is imposed for the small target image to remove the nontarget points while better highlighting the small target. For higher computational efficiency, an adapted version of the alternating direction method of multipliers is employed to solve the resulting minimization problem. Comparative experiments with synthetic and real data demonstrate that the proposed method is superior in detection performance to the state-of-the-art methods in terms of both objective measure and visual quality.