Martingale product estimators for sensitivity analysis in computational statistical physics
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 secondorder splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a GreenKubo 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 GreenKubo estimator. We compare on illustrative numerical tests the new estimators with results obtained by the GreenKubo method.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2112.00126
 Bibcode:
 2021arXiv211200126P
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematical Physics;
 Mathematics  Probability;
 65C05;
 65C20;
 65C40;
 60J27;
 60J75
 EPrint:
 34 pages, 4 figures