Long-Run Accuracy of Variational Integrators in the Stochastic Context

TL;DR

This paper analyzes the long-term statistical accuracy of a Lie-Trotter splitting method combining variational integrators and Ornstein-Uhlenbeck flows for inertial Langevin equations, showing it can accurately sample invariant measures.

Contribution

It introduces a Lie-Trotter splitting method for inertial Langevin equations and proves its invariant measure approximates the true measure with high accuracy, especially when energy errors are absent.

Findings

The splitting method accurately approximates the invariant measure.

Numerical validation confirms theoretical accuracy for various variational integrators.

The method achieves no error in sampling the invariant measure when energy is conserved.

Abstract

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.