Krylov Subspace Recycling With Randomized Sketching For Matrix Functions

TL;DR

This paper introduces a Krylov subspace recycling method enhanced with randomized sketching to efficiently compute sequences of matrix functions, reducing computational costs and enabling easier restarts.

Contribution

It presents a novel recycling algorithm that uses a closed-form approximation and randomized sketching, improving efficiency and restartability over previous methods.

Findings

The method reduces computational cost compared to traditional approaches.

Numerical experiments demonstrate improved efficiency and accuracy.

The approach enables easier restarting of recycling algorithms.

Abstract

A Krylov subspace recycling method for the efficient evaluation of a sequence of matrix functions acting on a set of vectors is developed. The method improves over the recycling methods presented in [Burke et al., arXiv:2209.14163, 2022] in that it uses a closed-form expression for the augmented FOM approximants and hence circumvents the use of numerical quadrature. We further extend our method to use randomized sketching in order to avoid the arithmetic cost of orthogonalizing a full Krylov basis, offering an attractive solution to the fact that recycling algorithms built from shifted augmented FOM cannot easily be restarted. The efficacy of the proposed algorithms is demonstrated with numerical experiments.