Parallel Matrix Function Evaluation via Initial value ODE modelling
For numerical linear algebra practitioners, it offers a new parallel approach to matrix function evaluation, but the results are preliminary and lack concrete performance comparisons.
The paper proposes ODE-based parallel algorithms (e.g., parareal) to evaluate matrix functions f(A) by modeling them as solutions of time-dependent equations, with numerical illustrations showing feasibility.
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.