Approximation of functions of large matrices with Kronecker structure

Abstract

We consider the numerical approximation of $f(\mathcal{A})b$ where $b\in {\mathbb{R}}^N$ and $\mathcal{A}$ is the sum of Kronecker products, that is $\mathcal{A}=M_2\otimes I+I\otimes M_1\in {\mathbb{R}}^{N\times N}$. Here $f$ is a regular function such that $f(\mathcal{A})$ 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.

References in corpus (1)

• Decay bounds for functions of matrices with banded or Kronecker structure

Cited by in corpus (2)

• Analytic functional calculus for two operators
• Local Exponential Methods: a domain decomposition approach to exponential time integration of PDEs