On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-01 DOI:10.1137/23m1586069

Ben S. Southworth, Thomas A. Manteuffel

{"title":"On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid","authors":"Ben S. Southworth, Thomas A. Manteuffel","doi":"10.1137/23m1586069","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1245-1258, September 2024. <br/> Abstract. The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction [math] will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning [math]. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to nonorthogonality, [math] may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the nonorthogonal correction, or the algorithm may diverge. This paper develops analytic formulae on how to construct “compatible” transfer operators in nonsymmetric AMG such that [math] in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between “ideal” transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Matrix Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1586069","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1245-1258, September 2024.
Abstract. The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction [math] will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning [math]. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to nonorthogonality, [math] may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the nonorthogonal correction, or the algorithm may diverge. This paper develops analytic formulae on how to construct “compatible” transfer operators in nonsymmetric AMG such that [math] in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between “ideal” transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.

查看原文本刊更多论文

论非对称代数多网格中的兼容转移算子

SIAM 矩阵分析与应用期刊》，第 45 卷，第 3 期，第 1245-1258 页，2024 年 9 月。摘要。有效代数多网格（AMG）算法的标准目标是开发能减弱互补误差模式的松弛和粗网格修正方案。在非对称环境下，粗网格校正[数学]几乎肯定在任何已知标准乘积中都是非正交（和发散）的，即[数学]。这就引入了一个新的考虑因素，即我们希望粗栅校正在适当的规范下尽可能接近正交。此外，由于非正交性，[math] 实际上可能会放大插值范围内的某些误差模式。因此，松弛不仅必须与插值相辅相成，还必须迅速消除非正交校正所放大的任何误差，否则算法可能会出现偏差。本文提出了如何在非对称 AMG 中构建 "兼容 "转移算子的解析公式，使得 [math] 在某种标准矩阵诱导规范中。本文还讨论了非对称设置中的不同规范选项、不同规范中 "理想 "转移算子之间的关系，以及对基于非对称还原的 AMG 收敛性的见解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.