An error control framework for computing the exponential of matrices arising from the finite element discretization

TL;DR

This paper develops an error control framework for computing the matrix exponential in finite element discretizations, focusing on matrices with specific structure, and demonstrates its effectiveness through numerical experiments.

Contribution

It introduces a novel approach using similarity transformations to estimate the numerical range, enabling accurate exponential computations for structured matrices.

Findings

Error bounds can be effectively estimated using similarity transformations.

The proposed method achieves prescribed accuracy in numerical experiments.

The framework is applicable to matrices with well-conditioned symmetric positive definite mass matrices.

Abstract

Several methods for computing the action of the matrix exponential $\mathrm{e}^{\boldsymbol{A}} \boldsymbol{b}$ are expressed by substituting $\boldsymbol{A}$ into a rational approximation of the scalar exponential function. The error of such methods can be estimated using the numerical range of $\boldsymbol{A}$, which enables the computation of $\mathrm{e}^{\boldsymbol{A}}\boldsymbol{b}$ with a prescribed accuracy. However, when the input matrix has the structure $\boldsymbol{A} = \tau \boldsymbol{M}^{-1} \boldsymbol{K}$, this approach is challenging because computing the bounding box of numerical range is difficult and the numerical range may be too large to construct rational approximations on it. In this paper, focusing on the case where $\boldsymbol{M}$ is a well-conditioned symmetric positive definite matrix, we propose considering the numerical range of a similarity transformed matrix of $\boldsymbol{A}$. The numerical range of transformed matrix is not only numerically computable but can also be theoretically bounded depending on properties of $\boldsymbol{K}$. Numerical experiments confirm that the computations can be performed within the prescribed error tolerance.