Orthogonal block Kaczmarz inner-iteration preconditioned flexible GMRES method for large-scale linear systems

IF 2.9 2区 数学 Q1 MATHEMATICS, APPLIED
Xin-Fang Zhang , Meng-Long Xiao , Zhuo-Heng He
{"title":"Orthogonal block Kaczmarz inner-iteration preconditioned flexible GMRES method for large-scale linear systems","authors":"Xin-Fang Zhang ,&nbsp;Meng-Long Xiao ,&nbsp;Zhuo-Heng He","doi":"10.1016/j.aml.2025.109529","DOIUrl":null,"url":null,"abstract":"<div><div>Kacamarz-type inner-iteration preconditioned flexible GMRES method, which was proposed by Du et al. (2021), is attractive for solving consistent linear systems. However, its inner iteration was only performed by several commonly used Kaczmarz-type methods, and required computing <span><math><mrow><mi>A</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msup></mrow></math></span> in advance, which is unfavorable for big data problems. To overcome these difficulties, we first propose a simple orthogonal block Kaczmarz method, based on the orthogonal block idea without preconditioning, which converges much faster than the mentioned-above Kaczmarz-type solvers. We then derive a simple orthogonal block Kaczmarz inner-iteration preconditioned flexible GMRES method, based on the orthogonal block Kaczmarz inner-iteration as a preconditioner, which is appealing for large-scale linear systems. The convergence analysis of which is also established. Finally, we provide some numerical examples to illustrate the effectiveness of the proposed methods compared with some state-of-the-art Kaczmarz-type methods.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"166 ","pages":"Article 109529"},"PeriodicalIF":2.9000,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965925000795","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

Kacamarz-type inner-iteration preconditioned flexible GMRES method, which was proposed by Du et al. (2021), is attractive for solving consistent linear systems. However, its inner iteration was only performed by several commonly used Kaczmarz-type methods, and required computing AAT in advance, which is unfavorable for big data problems. To overcome these difficulties, we first propose a simple orthogonal block Kaczmarz method, based on the orthogonal block idea without preconditioning, which converges much faster than the mentioned-above Kaczmarz-type solvers. We then derive a simple orthogonal block Kaczmarz inner-iteration preconditioned flexible GMRES method, based on the orthogonal block Kaczmarz inner-iteration as a preconditioner, which is appealing for large-scale linear systems. The convergence analysis of which is also established. Finally, we provide some numerical examples to illustrate the effectiveness of the proposed methods compared with some state-of-the-art Kaczmarz-type methods.
大型线性系统的正交块Kaczmarz内迭代预条件柔性GMRES方法
Du等人(2021)提出的kacamarz型内迭代预置柔性GMRES方法对于求解一致线性系统具有吸引力。但其内部迭代仅采用几种常用的kaczmarz型方法进行,且需要提前计算AAT,不利于解决大数据问题。为了克服这些困难,我们首先提出了一种简单的正交块Kaczmarz方法,该方法基于正交块思想,不加预处理,收敛速度比上述Kaczmarz型求解器快得多。然后,我们基于正交块Kaczmarz内迭代作为预条件,推导了一种简单的正交块Kaczmarz内迭代预条件柔性GMRES方法,该方法适用于大规模线性系统。并对其收敛性进行了分析。最后,我们提供了一些数值例子来说明所提出的方法与一些最先进的kaczmarz型方法的有效性。
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
Applied Mathematics Letters
Applied Mathematics Letters 数学-应用数学
CiteScore
7.70
自引率
5.40%
发文量
347
审稿时长
10 days
期刊介绍: The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.
×
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