A residual concept for Krylov subspace evaluation of the $\varphi$ matrix function

TL;DR

This paper introduces a new Krylov subspace algorithm with a residual-based stopping criterion and restarting procedure for efficiently computing the $oldsymbol{\varphi}$ matrix function, crucial in large-scale exponential integrators and network analysis.

Contribution

The paper presents a novel residual-based Krylov subspace method with a restart strategy for computing the $oldsymbol{\varphi}$ matrix function, ensuring convergence for matrices with stable numerical range.

Findings

The algorithm is efficient for large-scale evolution problems.

Numerical tests confirm the method's effectiveness on PDE discretizations.

Guaranteed convergence for matrices with stable numerical range.

Abstract

An efficient Krylov subspace algorithm for computing actions of the $\varphi$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. For matrices with numerical range in the stable complex half plane, we analyze residual convergence and prove that the restarted method is guaranteed to converge for any Krylov subspace dimension. Numerical tests demonstrate efficiency of our approach for solving large scale evolution problems resulting from discretized in space time-dependent PDEs, in particular, diffusion and convection-diffusion problems.