A Milstein scheme for SPDEs

TL;DR

This paper extends the Milstein scheme to certain semilinear SPDEs with multiplicative noise, enabling more efficient simulations with fewer computational resources.

Contribution

It introduces an infinite dimensional commutativity condition and develops an efficient Milstein-type scheme for SPDEs satisfying this condition.

Findings

The scheme is computationally more efficient than previous methods.

Numerical results demonstrate significant computational savings.

The method applies to stochastic heat and reaction diffusion equations.

Abstract

This article studies an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite dimensional commutativity condition. In particular, a suitable infinite dimensional analog of Milstein's algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation and stochastic reaction diffusion equations showing signifficant computational savings.