Geometric Integrators and Computing Ergodic Limits

Abstract

For many applications, it is interesting to compute the mean of a given function with respect to the invariant law of the diffusion, i.e. the ergodic limit. To evaluate these mean values, one often has to integrate a system over comparatively long time intervals. Geometric integrators considered in this chapter demonstrate computational superiority over long time intervals in comparison with standard schemes for SDEs.

In this chapter specific methods for two important classes of stochastic systems are constructed: stochastic Hamiltonian systems and Langevin-type equations. Symplectic methods for stochastic Hamiltonian systems proposed in the first part of this chapter have significant advantages over standard schemes for SDEs. The second part of the chapter presents special numerical methods (we call them quasi-symplectic) for Langevin-type equations which have widespread occurrence in models from physics, chemistry, and biology and also in Bayesian statistics. They are a workhorse of molecular dynamics under constant temperature conditions. In the third part of the chapter geometric integration ideas are applied to such models as Langevin equations and stochastic gradient systems for rigid body dynamics and the stochastic Landau-Lifshitz equation. In the last section errors arising in computing ergodic limits are analysed. Both the ensemble averaging and time averaging approaches are considered.