Uncertainty quantification for generalized Langevin dynamics

TL;DR

This paper introduces efficient finite difference estimators for goal-oriented sensitivity analysis of the generalized Langevin equation, utilizing coupling techniques to reduce variance and enable analysis of complex parameter perturbations.

Contribution

It develops a variance-reducing coupling framework for finite difference sensitivity estimators applicable to GLE, including cases where traditional methods fail.

Findings

Optimal common random path coupling minimizes estimator variance.

Proposed estimators effectively analyze sensitivity in particle dynamics.

Method enables global sensitivity analysis and non-local parameter perturbations.

Abstract

We present efficient finite difference estimators for goal-oriented sensitivity indices with applications to the generalized Langevin equation (GLE). In particular, we apply these estimators to analyze an extended variable formulation of the GLE where other well known sensitivity analysis techniques such as the likelihood ratio method are not applicable to key parameters of interest. These easily implemented estimators are formed by coupling the nominal and perturbed dynamics appearing in the finite difference through a common driving noise, or common random path. After developing a general framework for variance reduction via coupling, we demonstrate the optimality of the common random path coupling in the sense that it produces a minimal variance surrogate for the difference estimator relative to sampling dynamics driven by independent paths. In order to build intuition for the common random path coupling, we evaluate the efficiency of the proposed estimators for a comprehensive set of examples of interest in particle dynamics. These reduced variance difference estimators are also a useful tool for performing global sensitivity analysis and for investigating non-local perturbations of parameters, such as increasing the number of Prony modes active in an extended variable GLE.