Parareal computation of stochastic differential equations with time-scale separation: a numerical study

TL;DR

This paper explores a parallel-in-time algorithm combining microscopic stochastic models and macroscopic effective dynamics to efficiently solve stochastic differential equations with time-scale separation, demonstrating the importance of updating methods.

Contribution

It introduces a micro-macro parareal algorithm for stochastic differential equations and analyzes how different coupling and updating strategies affect efficiency.

Findings

Coarse-level updating via quantile functions improves performance.

Microscopic state generation is less critical if a suitable prior is used.

The method accelerates solutions by parallelizing across time for slow-fast stochastic systems.

Abstract

The parareal algorithm is known to allow for a significant reduction in wall clock time for accurate numerical solutions by parallelising across the time dimension. We present and test a micro-macro version of parareal, in which the fine propagator is based on a (high-dimensional, slow-fast) stochastic microscopic model, and the coarse propagator is based on a low-dimensional approximate effective dynamics at slow time scales. At the microscopic level, we use an ensemble of Monte Carlo particles, whereas the approximate coarse propagator uses the (deterministic) Fokker-Planck equation for the slow degrees of freedom. The required coupling between microscopic and macroscopic representations of the system introduces several design options, specifically on how to generate a microscopic probability distribution consistent with a required macroscopic probability distribution and how to perform the coarse-level updating of the macroscopic probability distribution in a meaningful manner. We numerically study how these design options affect the efficiency of the algorithm in a number of situations. The choice of the coarse-level updating operator strongly impacts the result, with a superior performance if addition and subtraction of the quantile function (inverse cumulative distribution) is used. How microscopic states are generated has a less pronounced impact, provided a suitable prior microscopic state is used.