Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs

TL;DR

This paper introduces a parallel-in-time Parareal method combining Monte Carlo simulations and ODE moment models to efficiently solve scalar McKean-Vlasov SDEs, including bimodal cases, with rapid convergence and good scalability.

Contribution

It develops a novel micro-macro Parareal algorithm that integrates Monte Carlo particle simulations with ODE-based coarse predictors for scalar McKean-Vlasov SDEs, handling bimodal distributions effectively.

Findings

Convergence occurs in few iterations depending on predictor quality.

The method achieves parallel speedup using a cheap ODE coarse propagator.

Numerical experiments confirm good weak scaling and accuracy.

Abstract

We propose a micro-macro parallel-in-time Parareal method for scalar McKean-Vlasov stochastic differential equations (SDEs). In the algorithm, the fine Parareal propagator is a Monte Carlo simulation of an ensemble of particles, while an approximate ordinary differential equation (ODE) description of the mean and the variance of the particle distribution is used as a coarse Parareal propagator to achieve speedup. We analyse the convergence behaviour of our method for a linear problem and provide numerical experiments indicating the parallel weak scaling of the algorithm on a set of examples. We show, with numerical experiments, that convergence typically takes place in a low number of iterations, depending on the quality of the ODE predictor. For bimodal SDEs, we avoid quality deterioration of the coarse predictor (compared to unimodal SDEs) through the usage of multiple ODEs, each describing the mean and variance of the particle distribution in locally unimodal regions of the phase space. The benefit of the proposed algorithm can be viewed through two lenses: (i) through the parallel-in-time lens, speedup is obtained through the use of a very cheap coarse integrator (an ODE moment model), and (ii) through the moment models lens, accuracy is iteratively gained through the use of parallel machinery as a corrector. In contrast to the isolated use of a moment model, the proposed method (iteratively) converges to the true distribution generated by the SDE.