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Fredholm integral equations with non-smooth kernels in weighted spaces: Nyström approximations, stability and convergence 加权空间中具有非光滑核的弗雷德霍姆积分方程：尼斯特伦近似、稳定性和收敛性
IF 1.7 2区 数学
Calcolo Pub Date : 2024-06-19 DOI: 10.1007/s10092-024-00593-1
Chafik Allouch
{"title":"Fredholm integral equations with non-smooth kernels in weighted spaces: Nyström approximations, stability and convergence","authors":"Chafik Allouch","doi":"10.1007/s10092-024-00593-1","DOIUrl":"https://doi.org/10.1007/s10092-024-00593-1","url":null,"abstract":"<p>This paper deals with the numerical solution of <i>Fredholm</i> integral equations of the second kind with endpoint algebraic singularities and with a kernel of <i>Green</i>’s function type. Novel <i>Nyström</i> schemes employing the <i>Gauss</i> quadrature rule are presented. These methods take into account the lack of smoothness along the diagonal of the kernel and may recover the full convergence rate of smooth kernels. A complete analysis of the stability and convergence is provided, and several numerical tests that illustrate the efficiency and accuracy of various approaches are considered.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
When rational functions meet virtual elements: the lightning virtual element method 当有理函数遇到虚元：闪电虚元法
IF 1.7 2区 数学
Calcolo Pub Date : 2024-06-13 DOI: 10.1007/s10092-024-00585-1
Manuel Trezzi, Umberto Zerbinati
{"title":"When rational functions meet virtual elements: the lightning virtual element method","authors":"Manuel Trezzi, Umberto Zerbinati","doi":"10.1007/s10092-024-00585-1","DOIUrl":"https://doi.org/10.1007/s10092-024-00585-1","url":null,"abstract":"<p>We propose a lightning Virtual Element Method that eliminates the stabilisation term by actually computing the virtual component of the local VEM basis functions using a lightning approximation. In particular, the lightning VEM approximates the virtual part of the basis functions using rational functions with poles clustered exponentially close to the corners of each element of the polygonal tessellation. This results in two great advantages. First, the mathematical analysis of a priori error estimates is much easier and essentially identical to the one for any other non-conforming Galerkin discretisation. Second, the fact that the lightning VEM truly computes the basis functions allows the user to access the point-wise value of the numerical solution without needing any reconstruction techniques. The cost of the local construction of the VEM basis is the implementation price that one has to pay for the advantages of the lightning VEM method, but the embarrassingly parallelizable nature of this operation will ultimately result in a cost-efficient scheme almost comparable to standard VEM and FEM.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical bounds on the Crouzeix ratio for a class of matrices 一类矩阵的 Crouzeix 比率数值界限
IF 1.7 2区 数学
Calcolo Pub Date : 2024-06-08 DOI: 10.1007/s10092-024-00580-6
Michel Crouzeix, Anne Greenbaum, Kenan Li
{"title":"Numerical bounds on the Crouzeix ratio for a class of matrices","authors":"Michel Crouzeix, Anne Greenbaum, Kenan Li","doi":"10.1007/s10092-024-00580-6","DOIUrl":"https://doi.org/10.1007/s10092-024-00580-6","url":null,"abstract":"<p>We provide numerical bounds on the Crouzeix ratio for KMS matrices <i>A</i> which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials <i>p</i> of the spectral norm of <i>p</i>(<i>A</i>) divided by the maximum absolute value of <i>p</i> on the numerical range of <i>A</i>. Our bounds satisfy the conjecture that this ratio is less than or equal to 2. We also give a precise description of these numerical ranges.\u0000</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"1849 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A block quaternion GMRES method and its convergence analysis 分块四元数 GMRES 方法及其收敛性分析
IF 1.7 2区 数学
Calcolo Pub Date : 2024-06-08 DOI: 10.1007/s10092-024-00576-2
Sinem Şimşek
{"title":"A block quaternion GMRES method and its convergence analysis","authors":"Sinem Şimşek","doi":"10.1007/s10092-024-00576-2","DOIUrl":"https://doi.org/10.1007/s10092-024-00576-2","url":null,"abstract":"<p>We consider the quaternion linear system <span>(AX = B)</span> for the unknown matrix <i>X</i>, where <i>A</i>, <i>B</i> are given <span>(ntimes n)</span>, <span>(ntimes s)</span> matrices with quaternion entries, motivated by applications that arise from fields such as quantum mechanics and signal processing. Our primary concern is the large-scale setting when <i>n</i> is large so that direct solutions are not feasible. We describe a block Krylov subspace method for the iterative solution of these quaternion linear systems. One difference compared to usual block Krylov subspace methods over complex Euclidean spaces is that the multiplication of quaternion scalars is not commutative. We describe a block quaternion Arnoldi process, taking noncommutativity features of quaternions into account, to generate an orthonormal basis for the quaternion Krylov space <span>(text {blockspan} { R_0, A R_0, dots , A^k R_0 })</span>, where <span>(R_0 = B - A X_0)</span> and <span>(X_0)</span> is an initial guess for the solution. Then the best solution of <span>(AX = B)</span> in the least-squares sense is sought in the generated Krylov space. We explain how these least-squares problems over quaternion Krylov spaces can be solved efficiently by means of Householder reflectors. Most notably, we analyze rigorously the convergence of the proposed block quaternion GMRES approach when <i>A</i> is diagonalizable, and in the more general setting when <i>A</i> is not necessarily diagonalizable by making use of the Jordan form of <i>A</i>. Finally, we report numerical results that confirm the validity of the deduced theoretical convergence results, in particular illustrate that the proposed block quaternion Krylov subspace method converges quickly when <i>A</i> has clustered eigenvalues.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On convergence of waveform relaxation for nonlinear systems of ordinary differential equations 论非线性常微分方程系统波形松弛的收敛性
IF 1.7 2区 数学
Calcolo Pub Date : 2024-06-05 DOI: 10.1007/s10092-024-00578-0
M. A. Botchev
{"title":"On convergence of waveform relaxation for nonlinear systems of ordinary differential equations","authors":"M. A. Botchev","doi":"10.1007/s10092-024-00578-0","DOIUrl":"https://doi.org/10.1007/s10092-024-00578-0","url":null,"abstract":"<p>To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard–Lindelöf iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, Liouville–Bratu–Gelfand, and nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"76 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hosvd-tmpe: an extrapolation method for multidimensional sequences Hosvd-tmpe：多维序列的外推法
IF 1.7 2区 数学
Calcolo Pub Date : 2024-06-03 DOI: 10.1007/s10092-024-00582-4
Abdeslem Hafid Bentbib, Khalid Jbilou, Ridwane Tahiri
{"title":"Hosvd-tmpe: an extrapolation method for multidimensional sequences","authors":"Abdeslem Hafid Bentbib, Khalid Jbilou, Ridwane Tahiri","doi":"10.1007/s10092-024-00582-4","DOIUrl":"https://doi.org/10.1007/s10092-024-00582-4","url":null,"abstract":"<p>Accelerating slowly convergent sequences is one of the main purposes of extrapolation methods. In this paper, we present a new tensor polynomial extrapolation method, which is based on a modified minimisation problem and some ideas leading to the recent Tensor Global Minimal Extrapolation Method (TG-MPE). We discuss the application of our method to fixed-point iterative process. An efficient algorithm via the higher order Singular Value Decomposition (HOSVD) is proposed for its implementation. The numerical tests show clearly the effectiveness and performance of the proposed method.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"48 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Greedy randomized sampling nonlinear Kaczmarz methods 贪婪随机抽样非线性卡兹马兹方法
IF 1.7 2区 数学
Calcolo Pub Date : 2024-05-13 DOI: 10.1007/s10092-024-00577-1
Yanjun Zhang, Hanyu Li, Ling Tang
{"title":"Greedy randomized sampling nonlinear Kaczmarz methods","authors":"Yanjun Zhang, Hanyu Li, Ling Tang","doi":"10.1007/s10092-024-00577-1","DOIUrl":"https://doi.org/10.1007/s10092-024-00577-1","url":null,"abstract":"<p>The nonlinear Kaczmarz method was recently proposed to solve the system of nonlinear equations. In this paper, we first discuss two greedy selection rules, i.e., the maximum residual and maximum distance rules, for the nonlinear Kaczmarz iteration. Then, based on them, two kinds of greedy randomized sampling methods are presented. Furthermore, we also devise four corresponding greedy randomized block methods, i.e., the multiple samples-based methods. The linear convergence in expectation of all the proposed methods is proved. Numerical results show that, in some applications, including brown almost linear function and generalized linear model, the greedy selection rules give faster convergence rates than the existing ones, and the block methods outperform the single sample-based ones.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility 通过人工可压缩性对随机纳维-斯托克斯方程进行数值逼近
IF 1.7 2区 数学
Calcolo Pub Date : 2024-04-06 DOI: 10.1007/s10092-024-00575-3
{"title":"Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility","authors":"","doi":"10.1007/s10092-024-00575-3","DOIUrl":"https://doi.org/10.1007/s10092-024-00575-3","url":null,"abstract":"<h3>Abstract</h3> <p>A constructive numerical approximation of the two-dimensional unsteady stochastic Navier–Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a penalty parameter <span> <span>(varepsilon )</span> </span>. Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the stochastic Navier–Stokes equations to occur within the originally introduced probability space.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"63 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weight calculation and convergence analysis of polyharmonic spline (PHS) with polynomials for different stencils 不同模版多项式多谐样条线（PHS）的权重计算和收敛性分析
IF 1.7 2区 数学
Calcolo Pub Date : 2024-04-05 DOI: 10.1007/s10092-024-00570-8
{"title":"Weight calculation and convergence analysis of polyharmonic spline (PHS) with polynomials for different stencils","authors":"","doi":"10.1007/s10092-024-00570-8","DOIUrl":"https://doi.org/10.1007/s10092-024-00570-8","url":null,"abstract":"<h3>Abstract</h3> <p>Recent developments in the field of the radial basis function-finite difference (RBF-FD) framework have been focused on conditionally positive definite polyharmonic splines (PHS). Within this context, our research focuses on deriving analytical weights for the RBF-FD+polynomials method within the framework of PHS. We provide convergence analyses for various stencils. To validate the accuracy of our derived formulations, we conduct a series of computational experiments across a range of test problems.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"52 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete projection methods for Fredholm–Hammerstein integral equations using Kumar and Sloan technique 使用 Kumar 和 Sloan 技术的 Fredholm-Hammerstein 积分方程离散投影法
IF 1.7 2区 数学
Calcolo Pub Date : 2024-03-26 DOI: 10.1007/s10092-024-00573-5
Ritu Nigam, Nilofar Nahid, Samiran Chakraborty, Gnaneshwar Nelakanti
{"title":"Discrete projection methods for Fredholm–Hammerstein integral equations using Kumar and Sloan technique","authors":"Ritu Nigam, Nilofar Nahid, Samiran Chakraborty, Gnaneshwar Nelakanti","doi":"10.1007/s10092-024-00573-5","DOIUrl":"https://doi.org/10.1007/s10092-024-00573-5","url":null,"abstract":"<p>The proposed work discusses discrete collocation and discrete Galerkin methods for second kind Fredholm–Hammerstein integral equations on half line <span>([0,infty ))</span> using Kumar and Sloan technique. In addition, the finite section approximation method is applied to transform the domain of integration from <span>([0, infty ))</span> to <span>([0,alpha ],~ alpha &gt;0)</span>. In contrast to previous studies in which the optimal order of convergence is achieved for projection methods, we attained superconvergence rates in uniform norm using piecewise polynomial basis function. Moreover, these superconvergence rates are further enhanced by using discrete multi-projection (collocation and Galerkin) methods. In order to support the provided theoretical framework, numerical examples are included as well.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"9 1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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