Mobility estimation for Langevin dynamics using control variates

TL;DR

This paper introduces a novel control variate-based variance reduction technique to efficiently estimate the mobility in two-dimensional Langevin dynamics, especially in the challenging low friction regime with non-separable potentials.

Contribution

It presents a new variance-reduction method for mobility estimation in Langevin dynamics, providing theoretical bounds and demonstrating effectiveness through numerical experiments.

Findings

The method reduces variance in mobility estimates.

Numerical results match previous evidence on low friction scaling.

Applicable to complex non-separable potentials.

Abstract

The scaling of the mobility of two-dimensional Langevin dynamics in a periodic potential as the friction vanishes is not well understood for non-separable potentials. Theoretical results are lacking, and numerical calculation of the mobility in the underdamped regime is challenging because the computational cost of standard Monte Carlo methods is inversely proportional to the friction coefficient, while deterministic methods are ill-conditioned. In this work, we propose a new variance-reduction method based on control variates for efficiently estimating the mobility of Langevin-type dynamics. We provide bounds on the bias and variance of the proposed estimator, and illustrate its efficacy through numerical experiments, first in simple one-dimensional settings and then for two-dimensional Langevin dynamics. Our results corroborate previous numerical evidence on the scaling of the mobility in the low friction regime for a simple non-separable potential.