Enhancing Block Cimmino for Sparse Linear Systems with Dense Columns via Schur Complement

F. S. Torun, Murat Manguoglu, C. Aykanat
{"title":"Enhancing Block Cimmino for Sparse Linear Systems with Dense Columns via Schur Complement","authors":"F. S. Torun, Murat Manguoglu, C. Aykanat","doi":"10.1137/21m1453475","DOIUrl":null,"url":null,"abstract":". The block Cimmino is a parallel hybrid row-block projection iterative method suc-4 cessfully used for solving general sparse linear systems. However, the convergence of the method 5 degrades when angles between subspaces spanned by the row-blocks are far from being orthogonal. 6 The density of columns as well as the numerical values of their nonzeros are more likely to contribute 7 to the non-orthogonality between row blocks. We propose a novel scheme to handle such “dense” 8 columns. The proposed scheme forms a reduced system by separating these columns and the re-9 spective rows from the original coefficient matrix and handling them via Schur complement. Then, 10 the angles between subspaces spanned by the row-blocks of the reduced system are expected to be 11 closer to orthogonal and the reduced system is solved efficiently by the block Conjugate Gradient 12 accelerated block Cimmino in fewer iterations. We also propose a novel metric for selecting “dense” 13 columns considering the numerical values. The proposed metric establishes an upper bound on the 14 sum of inner–products between row-blocks. Then, we propose an efficient algorithm for computing 15 the proposed metric for the columns. Extensive numerical experiments for a wide range of linear 16 systems confirm the effectiveness of the proposed scheme by achieving fewer iterations and faster 17 parallel solution time compared to the classical CG accelerated block Cimmino algorithm. 18","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"10 1","pages":"49-"},"PeriodicalIF":0.0000,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Sci. Comput.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/21m1453475","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

. The block Cimmino is a parallel hybrid row-block projection iterative method suc-4 cessfully used for solving general sparse linear systems. However, the convergence of the method 5 degrades when angles between subspaces spanned by the row-blocks are far from being orthogonal. 6 The density of columns as well as the numerical values of their nonzeros are more likely to contribute 7 to the non-orthogonality between row blocks. We propose a novel scheme to handle such “dense” 8 columns. The proposed scheme forms a reduced system by separating these columns and the re-9 spective rows from the original coefficient matrix and handling them via Schur complement. Then, 10 the angles between subspaces spanned by the row-blocks of the reduced system are expected to be 11 closer to orthogonal and the reduced system is solved efficiently by the block Conjugate Gradient 12 accelerated block Cimmino in fewer iterations. We also propose a novel metric for selecting “dense” 13 columns considering the numerical values. The proposed metric establishes an upper bound on the 14 sum of inner–products between row-blocks. Then, we propose an efficient algorithm for computing 15 the proposed metric for the columns. Extensive numerical experiments for a wide range of linear 16 systems confirm the effectiveness of the proposed scheme by achieving fewer iterations and faster 17 parallel solution time compared to the classical CG accelerated block Cimmino algorithm. 18
利用Schur补增强密集列稀疏线性系统的块逼近性
. 块Cimmino是一种并行混合行-块投影迭代方法,已成功地用于求解一般稀疏线性系统。然而,当由行块组成的子空间之间的角度远非正交时,方法5的收敛性会降低。列的密度及其非零值的数值更有可能导致行块之间的非正交性。我们提出了一种新颖的方案来处理这种“密集”的8列。该方案通过从原始系数矩阵中分离这些列和重新考虑的行,并通过舒尔补处理它们,形成了一个简化系统。然后,期望简化系统的行块所张成的子空间之间的夹角更接近于正交,并且使用块共轭梯度算法在更少的迭代中有效地求解了简化系统。我们还提出了一个考虑数值选择“密集”13列的新度量。所提出的度量建立了行块之间的内积和的上界。然后,我们提出了一种有效的算法来计算所建议的列度量。与经典的CG加速块Cimmino算法相比,该算法的迭代次数更少,并行求解时间更快,在广泛的线性系统中进行了大量数值实验,证实了该算法的有效性。18
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
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信