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The detection and correction of silent errors in pipelined Krylov subspace methods. 管道Krylov子空间方法中无声错误的检测与校正。
IF 2 3区 数学
Numerical Algorithms Pub Date : 2026-01-01 Epub Date: 2025-03-12 DOI: 10.1007/s11075-025-02037-5
Erin Claire Carson, Jakub Hercík
{"title":"The detection and correction of silent errors in pipelined Krylov subspace methods.","authors":"Erin Claire Carson, Jakub Hercík","doi":"10.1007/s11075-025-02037-5","DOIUrl":"https://doi.org/10.1007/s11075-025-02037-5","url":null,"abstract":"<p><p>As computational machines become larger and more complex, the probability of hardware failure rises. \"Silent errors\", or bit flips, may not be immediately apparent but can cause detrimental effects to algorithm behavior. In this work, we examine an algorithm-based approach to silent error detection in the context of pipelined Krylov subspace methods, in particular, Pipe-PR-CG, for the solution of linear systems. Our approach is based on using finite precision error analysis to bound the differences between quantities which should be equal in exact arithmetic. By monitoring select quantities during the iteration, we can detect when these bounds are violated, which indicates that a silent error has occurred. We use this approach to develop a fault-tolerant variant and also suggest a strategy for dynamically adapting the detection criteria. Our numerical experiments demonstrate the effectiveness of our approach.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"101 2","pages":"1175-1210"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12886330/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146168013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Column reduced digital nets. 列减少数字网。
IF 2 3区 数学
Numerical Algorithms Pub Date : 2026-01-01 Epub Date: 2025-03-24 DOI: 10.1007/s11075-025-02050-8
Vishnupriya Anupindi, Peter Kritzer
{"title":"Column reduced digital nets.","authors":"Vishnupriya Anupindi, Peter Kritzer","doi":"10.1007/s11075-025-02050-8","DOIUrl":"https://doi.org/10.1007/s11075-025-02050-8","url":null,"abstract":"<p><p>Digital nets provide an efficient way to generate integration nodes of quasi-Monte Carlo (QMC) rules. For certain applications, as e.g. in uncertainty quantification, we are interested in obtaining a speed-up in computing products of a matrix with the vectors corresponding to the nodes of a QMC rule. In the recent paper <i>The fast reduced QMC matrix-vector product</i> (Dick et al. J. Comput. Appl. Math. <b>440</b>, 115642 2024), a speed up was obtained by using so-called reduced lattices and row reduced digital nets. In this work, we propose a different multiplication algorithm where we exploit the repetitive structure of column reduced digital nets instead of row reduced digital nets. This method has advantages over the previous one, as it facilitates the error analysis when using the integration nodes in a QMC rule. We also provide an upper bound for the quality parameter of column reduced digital nets, and numerical tests to illustrate the efficiency of the new algorithm.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"101 3","pages":"1451-1473"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12945907/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147328194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Vem++, a C++ library to handle and play with the virtual element method. 一个c++库，用于处理和使用虚元素方法。
IF 2 3区 数学
Numerical Algorithms Pub Date : 2026-01-01 Epub Date: 2025-03-29 DOI: 10.1007/s11075-025-02059-z
Franco Dassi
{"title":"Vem++, a C++ library to handle and play with the virtual element method.","authors":"Franco Dassi","doi":"10.1007/s11075-025-02059-z","DOIUrl":"https://doi.org/10.1007/s11075-025-02059-z","url":null,"abstract":"<p><p>The Virtual Element Method (VEM) is an extension of the Finite Element Method (FEM) to handle polytopal meshes. After giving a short introduction of the VEM for a two dimensional Laplacian problem, we show the differences between an implementation of a VEM and a FEM code highlighting which are the main issues associated with the VEM framework. Furthermore, this paper will show one of the possible ways to face such issues: Vem++ a C++ library developed to \"deal and play\" with the VEM discretisation. This C++ library deals with the VEM, since there are several partial differential equations in two/three dimensions coming from both academic and engineering problems. Then, one can \"play\" with the VEM, since Vem++ has been designed so that one can plug-in new features such as new polytopes quadrature rules, new solvers and new virtual element spaces in a smart way.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"101 3","pages":"1633-1675"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12945964/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147328254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding. 联合特征值的随机计算方法及其在多参数特征值问题和寻根中的应用。
IF 2 3区 数学
Numerical Algorithms Pub Date : 2025-01-01 Epub Date: 2024-10-31 DOI: 10.1007/s11075-024-01971-0
Haoze He, Daniel Kressner, Bor Plestenjak
{"title":"Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding.","authors":"Haoze He, Daniel Kressner, Bor Plestenjak","doi":"10.1007/s11075-024-01971-0","DOIUrl":"https://doi.org/10.1007/s11075-024-01971-0","url":null,"abstract":"<p><p>It is well known that a family of <math><mrow><mi>n</mi> <mo>×</mo> <mi>n</mi></mrow> </math> commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the <i>n</i> joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"100 3","pages":"861-892"},"PeriodicalIF":2.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12515246/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145287745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimized Schwarz waveform relaxation methods for wave-heat coupling in one dimensional bounded domains. 一维有界域波热耦合的优化Schwarz波形松弛方法。
IF 2 3区 数学
Numerical Algorithms Pub Date : 2025-01-01 Epub Date: 2025-05-27 DOI: 10.1007/s11075-025-02100-1
Franz Chouly, Martin J Gander, Véronique Martin
{"title":"Optimized Schwarz waveform relaxation methods for wave-heat coupling in one dimensional bounded domains.","authors":"Franz Chouly, Martin J Gander, Véronique Martin","doi":"10.1007/s11075-025-02100-1","DOIUrl":"https://doi.org/10.1007/s11075-025-02100-1","url":null,"abstract":"<p><p>We are interested in heterogeneous domain decomposition methods to couple partial differential equations in space-time. The coupling can be used to describe the exchange of heat or forces or both, and has important applications like fluid-structure or ocean-atmosphere coupling. Heterogeneous domain decomposition methods permit furthermore the reuse of existing codes which represent long term investments, a further great advantage in applications. We require that our method can use different and adaptive time steps for the different models, can be executed in parallel, is robust, and can use independent fast inner solvers. An ideal candidate is Optimized Schwarz Waveform Relaxation (OSWR) that can be used without overlap, which is important for the different physical models. We focus here on the model problem of coupling a heat and a wave equation in one spatial dimension, which we consider to be a minimal example of relevance, and our goal is to design and analyze transmission conditions such that OSWR converges as fast as possible. We propose two strategies, a first one where we optimize the transmission using one common parameter, and a second one where we use the wave characteristics of one subdomain to choose one parameter, and then optimize the other. We illustrate our results with numerical experiments.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"100 4","pages":"1739-1763"},"PeriodicalIF":2.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12644143/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145642802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical solution of nonclassical boundary value problems 非经典边界值问题的数值求解
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-09-19 DOI: 10.1007/s11075-024-01946-1
Paola Boito, Yuli Eidelman, Luca Gemignani
{"title":"Numerical solution of nonclassical boundary value problems","authors":"Paola Boito, Yuli Eidelman, Luca Gemignani","doi":"10.1007/s11075-024-01946-1","DOIUrl":"https://doi.org/10.1007/s11075-024-01946-1","url":null,"abstract":"<p>We provide a new approach to obtain solutions of linear differential problems set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"28 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 posteriori error estimates for the exponential midpoint method for linear and semilinear parabolic equations 线性和半线性抛物方程指数中点法的后验误差估计
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-09-19 DOI: 10.1007/s11075-024-01940-7
Xianfa Hu, Wansheng Wang, Mengli Mao, Jiliang Cao
{"title":"A posteriori error estimates for the exponential midpoint method for linear and semilinear parabolic equations","authors":"Xianfa Hu, Wansheng Wang, Mengli Mao, Jiliang Cao","doi":"10.1007/s11075-024-01940-7","DOIUrl":"https://doi.org/10.1007/s11075-024-01940-7","url":null,"abstract":"<p>In this paper, the a posteriori error estimates of the exponential midpoint method for time discretization are established for linear and semilinear parabolic equations. Using the exponential midpoint approximation defined by a continuous and piecewise linear interpolation of nodal values yields suboptimal order estimates. Based on the property of the entire function, we introduce a continuous and piecewise quadratic time reconstruction of the exponential midpoint method to derive optimal order estimates; the error bounds solely depend on the discretization parameters, the data of the problem, and the approximation of the entire function. Several numerical examples are implemented to illustrate the theoretical results.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Global convergence of Newton’s method for the regularized p-Stokes equations 正则化 p-Stokes 方程牛顿法的全局收敛性
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-09-17 DOI: 10.1007/s11075-024-01941-6
Niko Schmidt
{"title":"Global convergence of Newton’s method for the regularized p-Stokes equations","authors":"Niko Schmidt","doi":"10.1007/s11075-024-01941-6","DOIUrl":"https://doi.org/10.1007/s11075-024-01941-6","url":null,"abstract":"<p>The motion of glaciers can be simulated with the <span>(varvec{p})</span>-Stokes equations. Up to now, Newton’s method to solve these equations has been analyzed in finite-dimensional settings only. We analyze the problem in infinite dimensions to gain a new viewpoint. We do that by proving global convergence of the infinite-dimensional Newton’s method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We prove that the additional diffusion term only causes minor differences in the solution compared to the original <span>(varvec{p})</span>-Stokes equations under the assumption of some regularity. Finally, we test our algorithms on two experiments: A reformulation of the experiment ISMIP-HOM <span>(varvec{B})</span> without sliding and a block with sliding. For the former, the approximation of exact step sizes for the Picard iteration and exact step sizes and Armijo step sizes for Newton’s method are superior in the experiment compared to the Picard iteration. For the latter experiment, Newton’s method with Armijo step sizes needs many iterations until it converges fast to the solution. Thus, Newton’s method with approximately exact step sizes is better than Armijo step sizes in this experiment.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"47 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Communication in multiplex transportation networks 多路运输网络中的通信
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-09-17 DOI: 10.1007/s11075-024-01943-4
Silvia Noschese, Lothar Reichel
{"title":"Communication in multiplex transportation networks","authors":"Silvia Noschese, Lothar Reichel","doi":"10.1007/s11075-024-01943-4","DOIUrl":"https://doi.org/10.1007/s11075-024-01943-4","url":null,"abstract":"<p>Complex networks are made up of vertices and edges. The edges, which may be directed or undirected, are equipped with positive weights. Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices in distinct layers represent different kinds of objects. Multiplex networks are special vertex-aligned multilayer networks, in which vertices in distinct layers are identified with each other and inter-layer edges connect each vertex with its copy in other layers and have a fixed weight <span>(gamma &gt;0)</span> associated with the ease of communication between layers. This paper discusses two different approaches to analyze communication in a multiplex. One approach focuses on the multiplex global efficiency by using the multiplex path length matrix, the other approach considers the multiplex total communicability. The sensitivity of both the multiplex global efficiency and the multiplex total communicability to structural perturbations in the network is investigated to help to identify intra-layer edges that should be strengthened to enhance communicability.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Homogeneous multigrid method for hybridizable interior penalty method 可混合内部惩罚法的均质多网格方法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-09-17 DOI: 10.1007/s11075-024-01942-5
Peipei Lu, Juan Wang
{"title":"Homogeneous multigrid method for hybridizable interior penalty method","authors":"Peipei Lu, Juan Wang","doi":"10.1007/s11075-024-01942-5","DOIUrl":"https://doi.org/10.1007/s11075-024-01942-5","url":null,"abstract":"<p>In this paper, we present a rigorous convergence analysis of a homogeneous multigrid method for the hybridizable interior penalty (IP-H) method. In particular, we use the injection operator developed in (Lu et al. <i>SIAM J. Numer. Anal.</i> <b>60</b>, 2293–2317 2022). It is shown that when the penalization parameter is sufficiently large, our proposed multigrid method is optimal under the assumption of elliptic regularity. Numerical experiments are provided to confirm the theoretical results.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"49 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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