A Parareal Algorithm with Spectral Coarse Solver

TL;DR

This paper introduces a novel Parareal algorithm utilizing spectral coarse solvers derived from localized reduced basis methods, enabling highly parallelizable and efficient time integration with rigorous error bounds.

Contribution

It presents a new class of Parareal algorithms that employ spectral approximations of transfer operators using randomized SVDs, enhancing parallel efficiency and accuracy.

Findings

Spectral coarse solvers outperform traditional single-step coarse solvers.

The approach allows increased parallelism by trading global iterations for local solves.

Numerical experiments confirm significant performance improvements.

Abstract

We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from spectral approximations of the transfer operators mapping initial values for a given time interval to the solution at the end of the interval. By leveraging randomized singular value decompositions, these spectral approximations are obtained embarrassingly parallel by computing local fine solutions for random initial values. We show a priori and a posteriori error bounds in terms of the computed singular values of the transfer operators. Our numerical experiments demonstrate that our approach can significantly outperform Parareal with single-step coarse solvers. At the same time, it permits to further increase parallelism in Parareal by trading global iterations for a larger number of independent local solves.