{"title":"Orthogonal block Kaczmarz inner-iteration preconditioned flexible GMRES method for large-scale linear systems","authors":"Xin-Fang Zhang , Meng-Long Xiao , 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 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.
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.