Subspace method for multiparameter‐eigenvalue problems based on tensor‐train representations

Abstract

AbstractIn this article, we solve ‐parameter eigenvalue problems (EPs), with any natural number by representing the problem using tensor‐trains (TTs) and designing a method based on this format. EPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving EP are restricted to , due to the fact that, to the best of our knowledge, no available solvers exist for and reasonable size of the involved matrices. In this article, we prove that computing the eigenvalues of a EP can be recast into computing the eigenvalues of TT‐operators. We adapted the algorithm in9 for symmetric eigenvalue problems in TT‐format to an algorithm for solving generic EPs. This leads to a subspace method whose subspace dimension does not depend on , in contrast to other subspace methods for EPs. This allows us to tackle EPs with and reasonable size of the matrices. We provide theoretical results and report numerical experiments. The MATLAB code is publicly available.