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

TL;DR

This paper introduces a tensor-train based subspace method for solving multiparameter eigenvalue problems (mEPs) that can handle more than three parameters efficiently, overcoming previous size and parameter limitations.

Contribution

The paper develops a novel tensor-train based algorithm for mEPs, enabling solutions for problems with more than three parameters and large matrices, which was previously infeasible.

Findings

Method successfully solves mEPs with m > 3.

Subspace dimension remains independent of m.

Numerical experiments validate the approach.

Abstract

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.