High-order integrator for sampling the invariant distribution
of a class of parabolic SPDEs with additive space-time noise
C.-E. Bréhier and G. Vilmart
Abstract.
We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear SPDEs driven by an additive space-time noise.
Combined with a postprocessor, the new method is a modification with negligible overhead of the standard linearized implicit Euler-Maruyama method.
We first provide an analysis of the integrator when applied for SDEs (finite dimension), where we prove that the method has order $2$ for the approximation of the invariant distribution, instead of $1$.
We then perform a stability analysis of the integrator in the semilinear SPDE context, and we prove in a linear case that a higher order of convergence is achieved.
Numerical experiments, including the semilinear heat equation driven by space-time white noise, confirm the theoretical findings and illustrate the efficiency of the approach.
Key Words. stochastic partial differential equations, postprocessor, invariant measure,
ergodicity, space-time white noise.