Error Analysis of Modified Langevin Dynamics

TL;DR

This paper investigates a modified Langevin dynamics that reduces computational costs by freezing slow particles, analyzing its ergodic properties and the impact on statistical error through theoretical proofs and numerical simulations.

Contribution

It proves ergodicity of a non-hypoelliptic modified Langevin dynamics and analyzes how the asymptotic variance depends on the kinetic energy modification parameters.

Findings

Ergodicity is established despite non-hypoellipticity.

Asymptotic variance increases with certain parameter choices.

Numerical results confirm theoretical analysis for various systems.

Abstract

We consider Langevin dynamics associated with a modified kinetic energy vanishing for small momenta. This allows us to freeze slow particles, and hence avoid the re-computation of inter-particle forces, which leads to computational gains. On the other hand, the statistical error may increase since there are a priori more correlations in time. The aim of this work is first to prove the ergodicity of the modified Langevin dynamics (which fails to be hypoelliptic), and next to analyze how the asymptotic variance on ergodic averages depends on the parameters of the modified kinetic energy. Numerical results illustrate the approach, both for low-dimensional systems where we resort to a Galerkin approximation of the generator, and for more realistic systems using Monte Carlo simulations.