基于残差的代用超平面扩展 Kaczmarz 算法，用于大型最小二乘法问题

IF 1.4 2区数学 Q1 MATHEMATICS

Calcolo Pub Date : 2024-08-04 DOI:10.1007/s10092-024-00605-0

Ke Zhang, Xiang-Xiang Chen, Xiang-Long Jiang

{"title":"基于残差的代用超平面扩展 Kaczmarz 算法，用于大型最小二乘法问题","authors":"Ke Zhang, Xiang-Xiang Chen, Xiang-Long Jiang","doi":"10.1007/s10092-024-00605-0","DOIUrl":null,"url":null,"abstract":"<p>We present a simple yet efficient two-stage extended Kaczmarz-type algorithm for solving large least squares problem. During each stage, the current iterate is projected onto a surrogate hyperplane instead of a single one, yielding remarkable reduction in the number of iteration steps and computational time. We prove that the proposed algorithm converges to the unique least-norm least-squares solution with a convergence factor asymptotically smaller than that for some existing randomized extended Kaczmarz-type algorithms. Numerical examples show that the new algorithm outperforms several counterparts for various test problems.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A residual-based surrogate hyperplane extended Kaczmarz algorithm for large least squares problems\",\"authors\":\"Ke Zhang, Xiang-Xiang Chen, Xiang-Long Jiang\",\"doi\":\"10.1007/s10092-024-00605-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We present a simple yet efficient two-stage extended Kaczmarz-type algorithm for solving large least squares problem. During each stage, the current iterate is projected onto a surrogate hyperplane instead of a single one, yielding remarkable reduction in the number of iteration steps and computational time. We prove that the proposed algorithm converges to the unique least-norm least-squares solution with a convergence factor asymptotically smaller than that for some existing randomized extended Kaczmarz-type algorithms. Numerical examples show that the new algorithm outperforms several counterparts for various test problems.</p>\",\"PeriodicalId\":9522,\"journal\":{\"name\":\"Calcolo\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Calcolo\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10092-024-00605-0\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Calcolo","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10092-024-00605-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

我们提出了一种简单而高效的两阶段扩展 Kaczmarz 型算法，用于求解大最小二乘法问题。在每个阶段，当前迭代都会投影到一个代理超平面上，而不是单个超平面，从而显著减少了迭代步数和计算时间。我们证明了所提出的算法能收敛到唯一的最小正则最小二乘法解，其收敛因子在渐近上小于现有的一些随机扩展卡茨马兹型算法。数值示例表明，新算法在各种测试问题上的表现优于几种同行算法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

A residual-based surrogate hyperplane extended Kaczmarz algorithm for large least squares problems

查看原文本刊更多论文

A residual-based surrogate hyperplane extended Kaczmarz algorithm for large least squares problems

We present a simple yet efficient two-stage extended Kaczmarz-type algorithm for solving large least squares problem. During each stage, the current iterate is projected onto a surrogate hyperplane instead of a single one, yielding remarkable reduction in the number of iteration steps and computational time. We prove that the proposed algorithm converges to the unique least-norm least-squares solution with a convergence factor asymptotically smaller than that for some existing randomized extended Kaczmarz-type algorithms. Numerical examples show that the new algorithm outperforms several counterparts for various test problems.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Calcolo 数学-数学

CiteScore

2.40

自引率

11.80%

发文量

审稿时长

>12 weeks

期刊介绍： Calcolo is a quarterly of the Italian National Research Council, under the direction of the Institute for Informatics and Telematics in Pisa. Calcolo publishes original contributions in English on Numerical Analysis and its Applications, and on the Theory of Computation. The main focus of the journal is on Numerical Linear Algebra, Approximation Theory and its Applications, Numerical Solution of Differential and Integral Equations, Computational Complexity, Algorithmics, Mathematical Aspects of Computer Science, Optimization Theory. Expository papers will also appear from time to time as an introduction to emerging topics in one of the above mentioned fields. There will be a "Report" section, with abstracts of PhD Theses, news and reports from conferences and book reviews. All submissions will be carefully refereed.