KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

TL;DR

KIOPS is a new adaptive Krylov subspace algorithm that efficiently computes matrix exponential functions for large-scale problems in exponential integrators, outperforming existing methods.

Contribution

The paper introduces KIOPS, a novel adaptive Krylov method using incomplete orthogonalization for exponential integrators, suitable for large-scale problems with limited spectral information.

Findings

KIOPS outperforms the state-of-the-art phipm algorithm in numerical experiments.

The method reduces computational complexity of exponential integrators.

KIOPS effectively handles large-scale problems with minimal spectral data.

Abstract

This paper presents a new algorithm KIOPS for computing linear combinations of $\varphi$-functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori. We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspace. We also present a novel adaptive procedure that significantly reduces the computational complexity of exponential integrators. Our numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm phipm.