Numerical solution of kinetic SPDEs via stochastic Magnus expansion

TL;DR

This paper introduces a numerical method using stochastic Magnus expansion to efficiently solve stochastic PDEs with two spatial variables, demonstrating significant speed-ups over traditional schemes.

Contribution

The paper presents a novel application of the stochastic Magnus expansion for solving SPDEs, showing improved accuracy and computational efficiency over Euler-Maruyama.

Findings

Magnus scheme outperforms Euler-Maruyama in accuracy

Achieves 20 to 200 times speed-up using GPU

Validated on stochastic Langevin equation and variable coefficient cases

Abstract

In this paper, we show how the It\^o-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity. As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler-Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler-Maruyama scheme, depending on the accuracy target and the spatial resolution.