Computable upper error bounds for Krylov approximations to matrix exponentials and associated $\varphi$-functions

TL;DR

This paper introduces a rigorous, computable upper bound for the error in Krylov approximations of matrix exponentials and related functions, improving error estimation accuracy in time-stepping algorithms.

Contribution

It derives a new a posteriori error estimate based on the residual, proven to be a rigorous upper bound and asymptotically accurate as the time step approaches zero.

Findings

The error bound is computationally efficient within Krylov space.

The bound is asymptotically correct for small time steps.

Numerical experiments demonstrate the bound's accuracy and effectiveness.

Abstract

An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $\varphi$-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.