Postprocessed integrators for the high order integration of ergodic SDEs

TL;DR

This paper extends the effective order concept to stochastic systems, introducing postprocessed integrators that improve sampling accuracy of ergodic SDEs, demonstrated through modifications of the stochastic θ-method and numerical experiments.

Contribution

It develops a novel postprocessing technique for high order integrators in ergodic SDEs, enhancing sampling accuracy for invariant measures.

Findings

Postprocessors achieve order two accuracy.

Method improves efficiency in stiff ergodic systems.

Numerical results confirm versatility and effectiveness.

Abstract

The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor. We show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems. The approach is illustrated with modifications of the stochastic $\theta$-method applied to Brownian dynamics, where postprocessors achieving order two are introduced. Numerical experiments, including stiff ergodic systems, illustrate the efficiency and versatility of the approach.