大规模参数相关赫米矩阵特征问题的统一逼近

Mattia Manucci, Emre Mengi, Nicola Guglielmi
{"title":"大规模参数相关赫米矩阵特征问题的统一逼近","authors":"Mattia Manucci, Emre Mengi, Nicola Guglielmi","doi":"arxiv-2409.05791","DOIUrl":null,"url":null,"abstract":"We consider the approximation of the smallest eigenvalue of a large\nparameter-dependent Hermitian matrix over a continuum compact domain. Our\napproach is based on approximating the smallest eigenvalue by the one obtained\nby projecting the large matrix onto a suitable small subspace, a practice\nwidely employed in the literature. The projection subspaces are constructed\niteratively (to reduce the error of the approximation where it is large) with\nthe addition of the eigenvectors of the parameter-dependent matrix at the\nparameter values where a surrogate error is maximal. The surrogate error is the\ngap between the approximation and a lower bound for the smallest eigenvalue\nproposed in [Sirkovic and Kressner, SIAM J. Matrix Anal. Appl., 37(2), 2016].\nUnlike the classical approaches, such as the successive constraint method, that\nmaximize such surrogate errors over a discrete and finite set, we maximize the\nsurrogate error over the continuum of all permissible parameter values\nglobally. We put particular attention to the lower bound, which enables us to\nformally prove the global convergence of our framework both in\nfinite-dimensional and infinite-dimensional settings. In the second part, we\nfocus on the approximation of the smallest singular value of a large\nparameter-dependent matrix, in case it is non-Hermitian, and propose another\nsubspace framework to construct a small parameter-dependent non-Hermitian\nmatrix whose smallest singular value approximates the original large-scale\nsmallest singular value. We perform numerical experiments on synthetic\nexamples, as well as on real examples arising from parametric PDEs. The\nnumerical experiments show that the proposed techniques are able to drastically\nreduce the size of the large parameter-dependent matrix, while ensuring an\napproximation error for the smallest eigenvalue/singular value below the\nprescribed tolerance.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix\",\"authors\":\"Mattia Manucci, Emre Mengi, Nicola Guglielmi\",\"doi\":\"arxiv-2409.05791\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the approximation of the smallest eigenvalue of a large\\nparameter-dependent Hermitian matrix over a continuum compact domain. Our\\napproach is based on approximating the smallest eigenvalue by the one obtained\\nby projecting the large matrix onto a suitable small subspace, a practice\\nwidely employed in the literature. The projection subspaces are constructed\\niteratively (to reduce the error of the approximation where it is large) with\\nthe addition of the eigenvectors of the parameter-dependent matrix at the\\nparameter values where a surrogate error is maximal. The surrogate error is the\\ngap between the approximation and a lower bound for the smallest eigenvalue\\nproposed in [Sirkovic and Kressner, SIAM J. Matrix Anal. Appl., 37(2), 2016].\\nUnlike the classical approaches, such as the successive constraint method, that\\nmaximize such surrogate errors over a discrete and finite set, we maximize the\\nsurrogate error over the continuum of all permissible parameter values\\nglobally. We put particular attention to the lower bound, which enables us to\\nformally prove the global convergence of our framework both in\\nfinite-dimensional and infinite-dimensional settings. In the second part, we\\nfocus on the approximation of the smallest singular value of a large\\nparameter-dependent matrix, in case it is non-Hermitian, and propose another\\nsubspace framework to construct a small parameter-dependent non-Hermitian\\nmatrix whose smallest singular value approximates the original large-scale\\nsmallest singular value. We perform numerical experiments on synthetic\\nexamples, as well as on real examples arising from parametric PDEs. The\\nnumerical experiments show that the proposed techniques are able to drastically\\nreduce the size of the large parameter-dependent matrix, while ensuring an\\napproximation error for the smallest eigenvalue/singular value below the\\nprescribed tolerance.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"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-2409.05791\",\"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-2409.05791","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

我们考虑的是在连续紧凑域上逼近一个大参数相关赫米矩阵的最小特征值。我们的方法是通过将大矩阵投影到一个合适的小子空间来逼近最小特征值,这是文献中广泛采用的一种做法。投影子空间是通过添加参数相关矩阵在参数值处的特征向量来逐级构建的(以减少近似误差较大时的误差),参数相关矩阵在参数值处的代用误差最大。代理误差是近似值与[Sirkovic 和 Kressner,SIAM J. Matrix Anal. Appl.我们特别关注下界,这使我们能够正式证明我们的框架在无限维和无限维设置下的全局收敛性。在第二部分中,我们重点讨论了依赖大参数矩阵的最小奇异值的近似问题(如果它是非ermitian 矩阵),并提出了另一个子空间框架来构造依赖小参数的非ermitian 矩阵,其最小奇异值近似于原始的大尺度最小奇异值。我们对合成示例以及参数 PDEs 产生的实际示例进行了数值实验。数值实验结果表明,所提出的技术能够大幅减小与参数相关的大型矩阵的大小,同时确保最小特征值/奇异值的近似误差低于规定的容差。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix
We consider the approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix over a continuum compact domain. Our approach is based on approximating the smallest eigenvalue by the one obtained by projecting the large matrix onto a suitable small subspace, a practice widely employed in the literature. The projection subspaces are constructed iteratively (to reduce the error of the approximation where it is large) with the addition of the eigenvectors of the parameter-dependent matrix at the parameter values where a surrogate error is maximal. The surrogate error is the gap between the approximation and a lower bound for the smallest eigenvalue proposed in [Sirkovic and Kressner, SIAM J. Matrix Anal. Appl., 37(2), 2016]. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We put particular attention to the lower bound, which enables us to formally prove the global convergence of our framework both in finite-dimensional and infinite-dimensional settings. In the second part, we focus on the approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian, and propose another subspace framework to construct a small parameter-dependent non-Hermitian matrix whose smallest singular value approximates the original large-scale smallest singular value. We perform numerical experiments on synthetic examples, as well as on real examples arising from parametric PDEs. The numerical experiments show that the proposed techniques are able to drastically reduce the size of the large parameter-dependent matrix, while ensuring an approximation error for the smallest eigenvalue/singular value below the prescribed tolerance.
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