张量结构线性系统的随机草图 TT-GMRES

Alberto Bucci, Davide Palitta, Leonardo Robol
{"title":"张量结构线性系统的随机草图 TT-GMRES","authors":"Alberto Bucci, Davide Palitta, Leonardo Robol","doi":"arxiv-2409.09471","DOIUrl":null,"url":null,"abstract":"In the last decade, tensors have shown their potential as valuable tools for\nvarious tasks in numerical linear algebra. While most of the research has been\nfocusing on how to compress a given tensor in order to maintain information as\nwell as reducing the storage demand for its allocation, the solution of linear\ntensor equations is a less explored venue. Even if many of the routines\navailable in the literature are based on alternating minimization schemes\n(ALS), we pursue a different path and utilize Krylov methods instead. The use\nof Krylov methods in the tensor realm is not new. However, these routines often\nturn out to be rather expensive in terms of computational cost and ALS\nprocedures are preferred in practice. We enhance Krylov methods for linear\ntensor equations with a panel of diverse randomization-based strategies which\nremarkably increase the efficiency of these solvers making them competitive\nwith state-of-the-art ALS schemes. The up-to-date randomized approaches we\nemploy range from sketched Krylov methods with incomplete orthogonalization and\nstructured sketching transformations to streaming algorithms for tensor\nrounding. The promising performance of our new solver for linear tensor\nequations is demonstrated by many numerical results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Randomized sketched TT-GMRES for linear systems with tensor structure\",\"authors\":\"Alberto Bucci, Davide Palitta, Leonardo Robol\",\"doi\":\"arxiv-2409.09471\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the last decade, tensors have shown their potential as valuable tools for\\nvarious tasks in numerical linear algebra. While most of the research has been\\nfocusing on how to compress a given tensor in order to maintain information as\\nwell as reducing the storage demand for its allocation, the solution of linear\\ntensor equations is a less explored venue. Even if many of the routines\\navailable in the literature are based on alternating minimization schemes\\n(ALS), we pursue a different path and utilize Krylov methods instead. The use\\nof Krylov methods in the tensor realm is not new. However, these routines often\\nturn out to be rather expensive in terms of computational cost and ALS\\nprocedures are preferred in practice. We enhance Krylov methods for linear\\ntensor equations with a panel of diverse randomization-based strategies which\\nremarkably increase the efficiency of these solvers making them competitive\\nwith state-of-the-art ALS schemes. The up-to-date randomized approaches we\\nemploy range from sketched Krylov methods with incomplete orthogonalization and\\nstructured sketching transformations to streaming algorithms for tensor\\nrounding. The promising performance of our new solver for linear tensor\\nequations is demonstrated by many numerical results.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09471\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09471","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

在过去十年中,张量已经显示出其作为数值线性代数中各种任务的重要工具的潜力。虽然大部分研究都集中在如何压缩给定张量以保持信息以及减少其分配的存储需求上,但线性张量方程的求解是一个探索较少的领域。尽管文献中的许多例程都是基于交替最小化方案(ALS),但我们却另辟蹊径,采用了克雷洛夫方法。在张量领域使用克雷洛夫方法并不新鲜。然而,这些例程的计算成本往往相当昂贵,因此 ALS 程序在实践中更受青睐。我们通过一系列基于随机化的策略来增强线性张量方程的 Krylov 方法,这些策略显著提高了求解器的效率,使其与最先进的 ALS 方案相媲美。我们采用的最新随机化方法包括具有不完全正交化和结构化草图变换的草图 Krylov 方法,以及用于张量包围的流算法。许多数值结果表明,我们的新求解器对线性张弦具有良好的性能。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Randomized sketched TT-GMRES for linear systems with tensor structure
In the last decade, tensors have shown their potential as valuable tools for various tasks in numerical linear algebra. While most of the research has been focusing on how to compress a given tensor in order to maintain information as well as reducing the storage demand for its allocation, the solution of linear tensor equations is a less explored venue. Even if many of the routines available in the literature are based on alternating minimization schemes (ALS), we pursue a different path and utilize Krylov methods instead. The use of Krylov methods in the tensor realm is not new. However, these routines often turn out to be rather expensive in terms of computational cost and ALS procedures are preferred in practice. We enhance Krylov methods for linear tensor equations with a panel of diverse randomization-based strategies which remarkably increase the efficiency of these solvers making them competitive with state-of-the-art ALS schemes. The up-to-date randomized approaches we employ range from sketched Krylov methods with incomplete orthogonalization and structured sketching transformations to streaming algorithms for tensor rounding. The promising performance of our new solver for linear tensor equations is demonstrated by many numerical results.
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
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:481959085
Book学术官方微信