Long-Run Accuracy of Variational Integrators in the Stochastic Context

Details Long-Run Accuracy of Variational Integrators in the Stochastic Context Long-Run Accuracy of Variational Integrators in the Stochastic Context

2007-12-26

arxiv.org/abs/0712.4123v5

Mathematics

Numerical Analysis

30 pages

Scientific paper

10.1137/090758842

This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin without error. Numerical validation is provided using explicit variational integrators with first, second, and fourth order accuracy.

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