Convergence of the micro-macro Parareal Method for a Linear Scale-Separated Ornstein-Uhlenbeck SDE: extended version

TL;DR

This paper analyzes the convergence of a multiscale micro-macro Parareal method applied to a linear scale-separated Ornstein-Uhlenbeck SDE, demonstrating how model errors affect convergence through numerical experiments.

Contribution

It introduces and tests a multiscale Parareal method for SDEs, linking model error to convergence behavior with numerical validation.

Findings

Model error impacts convergence rate

Numerical experiments confirm theoretical predictions

Effective low-dimensional models improve efficiency

Abstract

Time-parallel methods can reduce the wall clock time required for the accurate numerical solution of differential equations by parallelizing across the time-dimension. In this paper, we present and test the convergence behavior of a multiscale, micro-macro version of a Parareal method for stochastic differential equations (SDEs). In our method, the fine propagator of the SDE is based on a high-dimensional slow-fast microscopic model; the coarse propagator is based on a model-reduced version of the latter, that captures the low-dimensional, effective dynamics at the slow time scales. We investigate how the model error of the approximate model influences the convergence of the micro-macro Parareal algorithm and we support our analysis with numerical experiments. This is an extended and corrected version of [Domain Decomposition Methods in Science and Engineering XXVII. DD 2022, vol 149 (2024), pp. 69-76, Bossuyt, I., Vandewalle, S., Samaey, G.].