Some Remarks on Preconditioning Molecular Dynamics

TL;DR

This paper investigates how preconditioning in overdamped Langevin algorithms affects convergence speed and variance reduction, supported by theoretical analysis and numerical experiments in molecular simulation models.

Contribution

It provides a detailed analysis of the impact of preconditioning on asymptotic variance and convergence in simple molecular simulation models.

Findings

Preconditioning can reduce asymptotic variance.

Preconditioning can accelerate convergence to equilibrium.

Theoretical results are validated by numerical simulations.

Abstract

We consider a Preconditioned Overdamped Langevin algorithm that does not alter the invariant distribution (up to controllable discretisation errors) and ask whether preconditioning improves the standard model in terms of reducing the asymptotic variance and of accelerating convergence to equilibrium. We present a detailed study of the dependence of the asymptotic variance on preconditioning in some elementary toy models related to molecular simulation. Our theoretical results are supported by numerical simulations.