On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problems

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• 作者：Daniel Kressner^a ; ^{daniel.kressner@epfl.ch" class="auth_mail" title="E-mail the corresponding author} ; André ; Uschmajew^b ; ^{uschmajew@ins.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A18 ; 15A69 ; 41A25 ; 41A46 ; 41A63 ; 65J10
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：493
• 期：Complete
• 页码：556-572
• 全文大小：395 K

文摘

Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems and eigenvalue problems on Hilbert spaces. Although this question is central to the success of all existing solvers based on low-rank tensor techniques, very few of the results available so far allow to draw meaningful conclusions for higher dimensions. In this work, we develop a constructive framework to study low-rank approximability. One major assumption is that the involved linear operator admits a low-rank representation with respect to the chosen tensor format, a property that is known to hold in a number of applications. Simple conditions, which are shown to hold for a fairly general problem class, guarantee that our derived low-rank truncation error estimates do not deteriorate as the dimensionality increases.