PDE的非线性压缩降基近似

IF 1.2 4区 化学 Q3 CHEMISTRY, MULTIDISCIPLINARY
Albert Cohen, Charbel Farhat, Yvon Maday, Agustin Somacal
{"title":"PDE的非线性压缩降基近似","authors":"Albert Cohen, Charbel Farhat, Yvon Maday, Agustin Somacal","doi":"10.5802/crmeca.191","DOIUrl":null,"url":null,"abstract":"Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case.","PeriodicalId":10566,"journal":{"name":"Comptes Rendus. Chimie","volume":"56 1","pages":"0"},"PeriodicalIF":1.2000,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Nonlinear compressive reduced basis approximation for PDE’s\",\"authors\":\"Albert Cohen, Charbel Farhat, Yvon Maday, Agustin Somacal\",\"doi\":\"10.5802/crmeca.191\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case.\",\"PeriodicalId\":10566,\"journal\":{\"name\":\"Comptes Rendus. Chimie\",\"volume\":\"56 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-09-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Comptes Rendus. Chimie\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5802/crmeca.191\",\"RegionNum\":4,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Comptes Rendus. Chimie","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/crmeca.191","RegionNum":4,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 5

摘要

线性模型约简技术设计离线低维子空间,为参数化偏微分方程的近似解量身定制,用于快速在线数值模拟。当解族具有快速衰减的karhunen - lo特征值或Kolmogorov宽度,反映有限维线性空间的近似性时,这些方法,如适当正交分解(POD)或约基(RB)方法是非常有效的。另一方面,当这些量有一个缓慢的衰减时,它们变得无效,特别是对于具有参数相关激波位置的双曲输运方程的解族。这项工作的目的是探索非线性模型约简的能力,以规避这种特殊情况。为此,我们首先描述了非线性宽度的特定概念,这些概念对于上述族具有实质上更快的衰减。然后,我们讨论了一种系统的方法,通过从线性简化模型近似的第一个坐标进行非线性重建来实现更好的性能,从而使我们能够保持相同的基于投影的模型简化的“经典”框架。我们对该方法进行了分析,并报告了一个简单但具有指导意义的单变量测试用例的性能。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Nonlinear compressive reduced basis approximation for PDE’s
Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case.
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
Comptes Rendus. Chimie
Comptes Rendus. Chimie 化学-化学综合
CiteScore
2.10
自引率
25.00%
发文量
89
审稿时长
3 months
期刊介绍: The Comptes Rendus - Chimie are a free-of-charge, open access and peer-reviewed electronic scientific journal publishing original research articles. It is one of seven journals published by the Académie des sciences. Its objective is to enable researchers to quickly share their work with the international scientific community. The Comptes Rendus - Chimie also publish journal articles, thematic issues and articles reflecting the history of the Académie des sciences and its current scientific activity.
×
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