Subspace method for multiparameter-eigenvalue problems based on tensor-train representations
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In this paper we solve m-parameter eigenvalue problems (mEPs), with m any natural number by representing the problem using Tensor-Trains (TT) and designing a method based on this format. mEPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving mEP 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 mEP can be recast into computing the eigenvalues of TT-operators. We adapted the algorithm in <cit.> for symmetric eigenvalue problems in TT-format to an algorithm for solving generic mEPs. This leads to a subspace method whose subspace dimension does not depend on m, in contrast to other subspace methods for mEPS. This allows us to tackle mEPs with m > 3 and reasonable size of the matrices. We provide theoretical results and report numerical experiments. The MATLAB code is publicly available.
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