论文标题
基于张量训练表示的多参数 - 元素价值问题的子空间方法
Subspace method for multiparameter-eigenvalue problems based on tensor-train representations
论文作者
论文摘要
在本文中，我们解决了$ M $ - 参数特征值问题（$ M $ EPS），并通过使用张量训练（TT）来表示问题，并使用$ m $任何自然数字，并根据此格式设计一种方法。 $ m $ EP通常是将变量分开应用于可分离边界价值问题时会出现的。通常，解决$ m $ ep的方法仅限于$ m = 3 $，因为据我们所知，$ m> 3 $的最佳求解器和相关矩阵的合理尺寸不存在。在本文中，我们证明，计算$ M $ EP的特征值可以重新铸造到计算TT-operators的特征值中。我们在\ cite {dolgov2014a}中调整了TT-Format中对称特征问题问题的算法，以求解通用$ M $ EPS的算法。这导致了一个子空间方法，其子空间尺寸不取决于$ m $，与其他$ M $ EPS的子空间方法相反。这使我们能够以$ M> 3 $和合理的矩阵来解决$ m $ EPS。我们提供理论结果并报告数值实验。 MATLAB代码公开可用。
In this paper we solve $m$-parameter eigenvalue problems ($m$EPs), with $m$ any natural number by representing the problem using Tensor-Trains (TT) and designing a method based on this format. $m$EPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving $m$EP are restricted to $m = 3$, due to the fact that, to the best of our knowledge, no available solvers exist for $m>3$ and reasonable size of the involved matrices. In this paper, we prove that computing the eigenvalues of a $m$EP can be recast into computing the eigenvalues of TT-operators. We adapted the algorithm in \cite{Dolgov2014a} for symmetric eigenvalue problems in TT-format to an algorithm for solving generic $m$EPs. This leads to a subspace method whose subspace dimension does not depend on $m$, in contrast to other subspace methods for $m$EPS. This allows us to tackle $m$EPs with $m > 3$ and reasonable size of the matrices. We provide theoretical results and report numerical experiments. The MATLAB code is publicly available.
