{"title":"关于求解张量方程组的贪婪随机化卡茨马兹型方法","authors":"","doi":"10.1016/j.aml.2024.109261","DOIUrl":null,"url":null,"abstract":"<div><p>For solving the system of tensor equations <span><math><mrow><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>b</mi></mrow></math></span>, where <span><math><mrow><mi>x,b</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> and <span><math><mi>A</mi></math></span> is an <span><math><mi>m</mi></math></span>-order <span><math><mi>n</mi></math></span>-dimensional real tensor, we introduce two greedy Kaczmarz-type methods: the tensor relaxed greedy randomized Kaczmarz algorithm and the accelerated tensor relaxed greedy Kaczmarz algorithm. The deterministic convergence analysis of both methods is given based on the local tangential cone condition. Numerical results demonstrate that the greedy Kaczmarz-type methods are more efficient than the randomized Kaczmarz-type methods, and the accelerated greedy version exhibits significant acceleration.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On greedy randomized Kaczmarz-type methods for solving the system of tensor equations\",\"authors\":\"\",\"doi\":\"10.1016/j.aml.2024.109261\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For solving the system of tensor equations <span><math><mrow><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>b</mi></mrow></math></span>, where <span><math><mrow><mi>x,b</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> and <span><math><mi>A</mi></math></span> is an <span><math><mi>m</mi></math></span>-order <span><math><mi>n</mi></math></span>-dimensional real tensor, we introduce two greedy Kaczmarz-type methods: the tensor relaxed greedy randomized Kaczmarz algorithm and the accelerated tensor relaxed greedy Kaczmarz algorithm. The deterministic convergence analysis of both methods is given based on the local tangential cone condition. Numerical results demonstrate that the greedy Kaczmarz-type methods are more efficient than the randomized Kaczmarz-type methods, and the accelerated greedy version exhibits significant acceleration.</p></div>\",\"PeriodicalId\":55497,\"journal\":{\"name\":\"Applied Mathematics Letters\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2024-08-08\",\"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/S0893965924002817\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965924002817","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
摘要
为了求解张量方程组 Axm-1=b(其中 x,b∈Rn 且 A 为 m 阶 n 维实张量),我们介绍了两种贪婪卡茨马兹型方法:张量松弛贪婪随机卡茨马兹算法和加速张量松弛贪婪卡茨马兹算法。基于局部切向锥条件,给出了这两种方法的确定性收敛分析。数值结果表明,贪心 Kaczmarz 型方法比随机 Kaczmarz 型方法更有效,而加速贪心版本则表现出显著的加速性。
On greedy randomized Kaczmarz-type methods for solving the system of tensor equations
For solving the system of tensor equations , where and is an -order -dimensional real tensor, we introduce two greedy Kaczmarz-type methods: the tensor relaxed greedy randomized Kaczmarz algorithm and the accelerated tensor relaxed greedy Kaczmarz algorithm. The deterministic convergence analysis of both methods is given based on the local tangential cone condition. Numerical results demonstrate that the greedy Kaczmarz-type methods are more efficient than the randomized Kaczmarz-type methods, and the accelerated greedy version exhibits significant acceleration.
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.