An accurate restarting for shift-and-invert Krylov subspaces computing matrix exponential actions of nonsymmetric matrices

TL;DR

This paper introduces an improved residual--time restarting method for the shift-and-invert Krylov subspace approach, enhancing accuracy and efficiency in computing matrix exponential actions of nonsymmetric matrices.

Contribution

It extends the RT restarting method to avoid accuracy loss and efficiently implements it with a single LU factorization in the SAI Krylov method.

Findings

Enhanced accuracy in matrix exponential computations.

Reduced computational cost with a single LU factorization.

Demonstrated efficiency improvements in numerical experiments.

Abstract

An accurate residual--time (AccuRT) restarting for computing matrix exponential actions of nonsymmetric matrices by the shift-and-invert (SAI) Krylov subspace method is proposed. The proposed restarting method is an extension of the recently proposed RT (residual--time) restarting and it is designed to avoid a possible accuracy loss in the conventional RT restarting. An expensive part of the SAI Krylov method is solution of linear systems with the shifted matrix. Since the AccuRT algorithm adjusts the shift value, we discuss how the proposed restarting can be implemented with just a single LU~factorization (or a preconditioner setup) of the shifted matrix. Numerical experiments demonstrate an improved accuracy and efficiency of the approach.