{"title":"对称正半有限线性系统的双网格收敛理论","authors":"Xuefeng Xu","doi":"arxiv-2409.09442","DOIUrl":null,"url":null,"abstract":"This paper is devoted to the convergence theory of two-grid methods for\nsymmetric positive semidefinite linear systems, with particular focus on the\nsingular case. In the case where the Moore--Penrose inverse of coarse-grid\nmatrix is used as a coarse solver, we derive a succinct identity for\ncharacterizing the convergence factor of two-grid methods. More generally, we\npresent some convergence estimates for two-grid methods with approximate coarse\nsolvers, including both linear and general cases. A key feature of our analysis\nis that it does not require any additional assumptions on the system matrix,\nespecially on its null space.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Two-grid convergence theory for symmetric positive semidefinite linear systems\",\"authors\":\"Xuefeng Xu\",\"doi\":\"arxiv-2409.09442\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper is devoted to the convergence theory of two-grid methods for\\nsymmetric positive semidefinite linear systems, with particular focus on the\\nsingular case. In the case where the Moore--Penrose inverse of coarse-grid\\nmatrix is used as a coarse solver, we derive a succinct identity for\\ncharacterizing the convergence factor of two-grid methods. More generally, we\\npresent some convergence estimates for two-grid methods with approximate coarse\\nsolvers, including both linear and general cases. A key feature of our analysis\\nis that it does not require any additional assumptions on the system matrix,\\nespecially on its null space.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09442\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09442","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Two-grid convergence theory for symmetric positive semidefinite linear systems
This paper is devoted to the convergence theory of two-grid methods for
symmetric positive semidefinite linear systems, with particular focus on the
singular case. In the case where the Moore--Penrose inverse of coarse-grid
matrix is used as a coarse solver, we derive a succinct identity for
characterizing the convergence factor of two-grid methods. More generally, we
present some convergence estimates for two-grid methods with approximate coarse
solvers, including both linear and general cases. A key feature of our analysis
is that it does not require any additional assumptions on the system matrix,
especially on its null space.