Martingale product estimators for sensitivity analysis in computational statistical physics

TL;DR

This paper introduces martingale product estimators for linear response in stochastic dynamics, providing a bias- and variance-controlled alternative to Green-Kubo estimators with applications in molecular transport property estimation.

Contribution

The paper develops a new class of estimators based on martingale products, extending likelihood ratio methods and analyzing their bias and variance in Langevin dynamics.

Findings

Variance of the new estimators is uniformly bounded in time.

The estimators outperform Green-Kubo in variance growth.

Numerical tests demonstrate the effectiveness of the estimators.

Abstract

We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name "martingale product estimators". We present a systematic derivation of the martingale product estimator, and show how to construct such estimator so its bias is consistent with the weak order of the numerical scheme that approximates the underlying stochastic differential equation. Motivated by the estimation of transport properties in molecular systems, we present a rigorous numerical analysis of the bias and variance for these new estimators in the case of Langevin dynamics. We prove that the variance is uniformly bounded in time and derive a specific form of the estimator for second-order splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a Green-Kubo estimator, motivated, in part, by its variance growing linearly in time. Presented analysis shows that the new martingale product estimators, having uniformly bounded variance in time, offer a competitive alternative to the traditional Green-Kubo estimator. We compare on illustrative numerical tests the new estimators with results obtained by the Green-Kubo method.