Improving dynamical properties of metropolized discretizations of overdamped Langevin dynamics

TL;DR

This paper proposes modifications to Metropolized discretizations of overdamped Langevin dynamics to enhance their dynamical properties, either by improving strong order or reducing bias in transport coefficient estimation.

Contribution

It introduces new modifications to standard Metropolis-Hastings corrections that improve dynamical accuracy and bias reduction in overdamped Langevin discretizations.

Findings

Enhanced strong order of numerical methods.

Reduced bias in transport coefficient estimation.

Improved dynamical consistency of the discretizations.

Abstract

The discretization of overdamped Langevin dynamics, through schemes such as the Euler-Maruyama method, can be corrected by some acceptance/rejection rule, based on a Metropolis-Hastings criterion for instance. In this case, the invariant measure sampled by the Markov chain is exactly the Boltzmann-Gibbs measure. However, rejections perturb the dynamical consistency of the resulting numerical method with the reference dynamics. We present in this work some modifications of the standard correction of discretizations of overdamped Langevin dynamics on compact spaces by a Metropolis-Hastings procedure, which allow us to either improve the strong order of the numerical method, or to decrease the bias in the estimation of transport coefficients characterizing the effective dynamical behavior of the dynamics. For the latter approach, we rely on modified numerical schemes together with a Barker rule for the acceptance/rejection criterion.