High-order integrator for sampling the invariant distribution of a class of parabolic SPDEs with additive space-time noise
For researchers in computational stochastic PDEs, this provides a more accurate method for sampling invariant distributions, though it is an incremental improvement over existing linearized implicit Euler-Maruyama.
The paper introduces a high-order time-integrator for sampling the invariant distribution of semilinear SPDEs with additive space-time noise, achieving order 2 accuracy for SDEs and higher order for linear SPDEs, with negligible computational overhead over the standard Euler-Maruyama method.
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.