Krylov subspace restarting for matrix Laplace transforms

TL;DR

This paper introduces a new stable restart algorithm for approximating matrix functions using Arnoldi methods, specifically for functions expressed as Laplace transforms, improving efficiency and stability over previous approaches.

Contribution

It develops a novel error representation for Arnoldi approximations of Laplace transform functions and extends restart algorithms from Stieltjes functions to general Laplace transforms.

Findings

The new algorithm is more stable and efficient.

Numerical experiments demonstrate improved performance.

Extension from Stieltjes to general Laplace transforms.

Abstract

A common way to approximate $F(A)b$ -- the action of a matrix function on a vector -- is to use the Arnoldi approximation. Since a new vector needs to be generated and stored in every iteration, one is often forced to rely on restart algorithms which are either not efficient, not stable or only applicable to restricted classes of functions. We present a new representation of the error of the Arnoldi iterates if the function $F$ is given as a Laplace transform. Based on this representation we build an efficient and stable restart algorithm. In doing so we extend earlier work for the class of Stieltjes functions which are special Laplace transforms. We report several numerical experiments including comparisons with the restart method for Stieltjes functions.