Approximation of functions of large matrices with Kronecker structure

TL;DR

This paper introduces an efficient numerical method for approximating functions of large matrices with Kronecker structure, reducing memory and computational costs, especially for the exponential function, demonstrated through numerical experiments.

Contribution

It presents a novel computational strategy that significantly decreases memory and computational requirements for approximating functions of Kronecker-structured matrices.

Findings

Reduced memory usage in approximations

Enhanced efficiency for exponential functions

Validated effectiveness through numerical experiments

Abstract

We consider the numerical approximation of $f({\cal A})b$ where $b\in{\mathbb R}^{N}$ and $\cal A$ is the sum of Kronecker products, that is ${\cal A}=M_2 \otimes I + I \otimes M_1\in{\mathbb R}^{N\times N}$. Here $f$ is a regular function such that $f({\cal A})$ is well defined. We derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations, with special emphasis on the exponential function, for which the new procedure becomes particularly advantageous. Our findings are illustrated by numerical experiments with typical functions used in applications.