×

Subspace method for multiparameter-eigenvalue problems based on tensor-train representations. (English) Zbl 07584147

Summary: In this article, we solve \(m\)-parameter eigenvalue problems (\(m\)EPs), with \(m\) any natural number by representing the problem using tensor-trains (TTs) 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 article, 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 [S. V. Dolgov et al., Comput. Phys. Commun. 185, No. 4, 1207–1216 (2014; Zbl 1344.65043)] 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.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15-04 Software, source code, etc. for problems pertaining to linear algebra

Citations:

Zbl 1344.65043
PDFBibTeX XMLCite
Full Text: DOI arXiv