Comparisons of the Parallel Preconditioners for Large Nonsymmetric Sparse Linear Systems on a Parallel Computer

Sangback Ma
{"title":"Comparisons of the Parallel Preconditioners for Large Nonsymmetric Sparse Linear Systems on a Parallel Computer","authors":"Sangback Ma","doi":"10.1142/S0129053304000232","DOIUrl":null,"url":null,"abstract":"In this paper we compare various parallel preconditioners for solving large sparse nonsymmetric linear systems. They are Block Jacobi, Point-SSOR, ILU(0) in the wavefront order, ILU(0) in the multi-color order, SPAI(SParse Approximate Inverse), and Multi-Color Block SOR. The Block Jacobi and Point-SSOR are well-known, and ILU(0) is one of the most popular preconditioners, but it is inherently serial. ILU(0) in the wavefront order maximizes the parallelism, and ILU(0) in the multi-color order achieves the parallelism of order (N), where N is the order of the matrix. The SPAI tries to capture the approximate inverse in sparse form, which, then, is expected to be a scalable preconditioner. Finally, we implemented the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a non-deteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024, and for an ill-conditioned matrix from the shell problem from the Harwell–Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR and Block Jacobi converges. Based on this we recommend that the Multi-Color Block SOR is the most robust preconditioner out of the preconditioners considered.","PeriodicalId":270006,"journal":{"name":"Int. J. High Speed Comput.","volume":"21 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2004-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Int. J. High Speed Comput.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/S0129053304000232","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5

Abstract

In this paper we compare various parallel preconditioners for solving large sparse nonsymmetric linear systems. They are Block Jacobi, Point-SSOR, ILU(0) in the wavefront order, ILU(0) in the multi-color order, SPAI(SParse Approximate Inverse), and Multi-Color Block SOR. The Block Jacobi and Point-SSOR are well-known, and ILU(0) is one of the most popular preconditioners, but it is inherently serial. ILU(0) in the wavefront order maximizes the parallelism, and ILU(0) in the multi-color order achieves the parallelism of order (N), where N is the order of the matrix. The SPAI tries to capture the approximate inverse in sparse form, which, then, is expected to be a scalable preconditioner. Finally, we implemented the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a non-deteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024, and for an ill-conditioned matrix from the shell problem from the Harwell–Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR and Block Jacobi converges. Based on this we recommend that the Multi-Color Block SOR is the most robust preconditioner out of the preconditioners considered.
大型非对称稀疏线性系统并行预调节器在并行计算机上的比较
本文比较了求解大型稀疏非对称线性系统的各种并行预调节器。分别是Block Jacobi、Point-SSOR、波前阶ILU(0)、多色阶ILU(0)、SPAI(SParse Approximate Inverse)和多色块SOR。Block Jacobi和Point-SSOR是众所周知的,ILU(0)是最流行的预调节器之一,但它本质上是串行的。波前阶的ILU(0)使并行度最大化,多色阶的ILU(0)实现了N阶的并行度,其中N为矩阵的阶数。SPAI试图以稀疏形式捕获近似逆,因此,它有望成为可扩展的预调节器。最后,我们结合直接稀疏矩阵求解器实现了多色块SOR预调节器。对于拉普拉斯矩阵,已知SOR方法在使用多色排序时具有非退化的收敛率。由于大部分时间都花在对角反转上,这是在每个处理器上完成的,我们期望它是一个很好的可扩展预调节器。最后,由于阻断作用,对病态问题会有效果。实验对两个问题的有限差分离散化进行了不同的网格大小变化到1024×1024,并对来自Harwell-Boeing收集的壳体问题的病态矩阵进行了实验。采用CRAY-T3E, 128个节点。MPI库用于进程间通信。结果表明,多色块SOR和多色排序的ILU(0)对于有限差分矩阵有最好的性能,对于壳问题只有多色块SOR和块Jacobi收敛。基于此,我们建议多色块SOR是所考虑的预调节器中最鲁棒的预调节器。
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:604180095
Book学术官方微信