Postprocessed integrators for the high order integration of ergodic SDEs
For researchers in stochastic simulation and molecular dynamics, this provides a new way to improve accuracy of long-time integration of ergodic SDEs, though the extension is incremental.
The paper extends the concept of effective order from deterministic to stochastic differential equations, constructing high-order integrators for sampling invariant measures of ergodic systems. Postprocessed modifications of the stochastic θ-method achieve order two, with numerical experiments demonstrating efficiency on stiff ergodic systems.
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 $θ$-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.