{"title":"Rounding-Error Analysis of Multigrid [math]-Cycles","authors":"Stephen F. McCormick, Rasmus Tamstorf","doi":"10.1137/23m1582898","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. <br/> Abstract. Earlier work on rounding-error analysis of multigrid was restricted to cycles that used one relaxation step before coarsening and none afterwards. The present paper extends this analysis to two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the coarse grid to allow for an approximate coarse-grid solve. Leveraging previously published results, this two-grid theory can then be extended to general [math]-cycles, as well as full multigrid. It can also be extended to mixed-precision iterative refinement based on these cycles. An added benefit of the theory here over previous work is that it is obtained in a more organized, transparent, and simpler way.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"122 1","pages":""},"PeriodicalIF":3.0000,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Scientific Computing","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1582898","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Scientific Computing, Ahead of Print. Abstract. Earlier work on rounding-error analysis of multigrid was restricted to cycles that used one relaxation step before coarsening and none afterwards. The present paper extends this analysis to two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the coarse grid to allow for an approximate coarse-grid solve. Leveraging previously published results, this two-grid theory can then be extended to general [math]-cycles, as well as full multigrid. It can also be extended to mixed-precision iterative refinement based on these cycles. An added benefit of the theory here over previous work is that it is obtained in a more organized, transparent, and simpler way.
期刊介绍：
The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems.
SISC papers are classified into three categories:
1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms.
2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist.
3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.
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