Error analysis of the transport properties of Metropolized schemes

TL;DR

This paper analyzes the discretization errors in computing transport coefficients for Brownian dynamics using Metropolis-adjusted Langevin algorithms, establishing an order one error in the time step and supporting findings with numerical simulations.

Contribution

It provides a rigorous analysis of the discretization error order in transport coefficient estimation for Metropolized schemes, which was previously not well-understood.

Findings

Error is of order one in the time step

Green-Kubo and Einstein formulas yield similar error orders

Numerical simulations confirm theoretical results

Abstract

We consider in this work the numerical computation of transport coefficients for Brownian dynamics. We investigate the discretization error arising when simulating the dynamics with the Smart MC algorithm (also known as Metropolis-adjusted Langevin algorithm). We prove that the error is of order one in the time step, when using either the Green-Kubo or the Einstein formula to estimate the transport coefficients. We illustrate our results with numerical simulations.