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Tensor-Krylov method for computing eigenvalues of parameter-dependent matrices by Koen Ruymbeek; Karl Meerbergen; Wim Michiels is a Mathematics article available to read on EtoBox.

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In this paper we extend the Residual Arnoldi method for calculating an extreme eigenvalue (e.g. largest real part, dominant, etc.) to the case where the matrices depend on parameters. The difference between this Arnoldi method and the classical Arnoldi algorithm is that in the former the residual is added to the subspace. We develop a Tensor-Krylov method that can be interpreted as an application of the Residual Arnoldi algorithm to a set of matrices obtained by evaluating a parameter-dependent matrix in parameter values on a grid. The subspace contains an approximate Krylov space for all these points. Instead of adding the residuals for all parameter values to the subspace we create a low-rank approximation of the matrix consisting of these residuals and add only the column space to the subspace. In order to keep the computations efficient, it is needed to limit the dimension of the subspace and to restart once the subspace has reached the prescribed maximal dimension. The novelty of this approach is twofold. Firstly, we observed that a large error in the low-rank approximations is allowed without slowing down the convergence, which implies that we can do more iterations before re

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Author

Koen Ruymbeek; Karl Meerbergen; Wim Michiels

Publisher

Elsevier BV

Published

2022

Language

EN

Field

Mathematics (Physical Sciences)