Parallel Matrix Function Evaluation via Initial value ODE modelling

TL;DR

This paper introduces ODE-based methods for efficiently computing matrix functions by modeling them as solutions to time-dependent equations and applying parallel algorithms like parareal, enabling faster evaluations especially for steady-state solutions.

Contribution

It presents a novel approach combining ODE modeling with parallel algorithms for matrix function evaluation, enhancing computational efficiency and scalability.

Findings

Effective parallel algorithms for matrix function evaluation.

Successful numerical illustrations demonstrating method viability.

Potential for improved computational speed in large-scale problems.

Abstract

The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite time $T$ of a time dependent equation. We use parallel algorithms, such as the parareal method, on the time interval $[0, T]$ in order to solve the evolution equation obtained. When $f(A)$ is reached as a stable steady state, it can be computed by combining parareal algorithms and optimal control techniques. Numerical illustrations are given.