Error bound analysis of the stochastic parareal algorithm

TL;DR

This paper analyzes the error bounds of the stochastic parareal algorithm, a probabilistic parallel-in-time method, providing theoretical guarantees and numerical validation for nonlinear ODEs.

Contribution

It derives superlinear and linear mean-square error bounds for SParareal applied to nonlinear ODEs, advancing understanding of its accuracy.

Findings

Error bounds match numerical experiments

Superlinear convergence observed in certain cases

Theoretical analysis applies to nonlinear systems

Abstract

Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.