A Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields

TL;DR

This paper presents a hybrid Monte Carlo method for efficiently sampling rare events in stochastic fields, validated on the Burgers equation, and applicable to high-order moments and instanton configurations.

Contribution

A novel HMC algorithm tailored for large-deviation statistics in stochastic hydrodynamics, enabling efficient sampling of rare events and high-order moments.

Findings

Successfully reproduces multiscale properties of Burgers equation

Enhances sampling of rare events by orders of magnitude

Explores statistical fluctuations around instanton configurations

Abstract

We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large-deviation statistics in stochastic hydrodynamics. Based on the path-integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multiscale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by orders of magnitudes, the probability to sample extreme and rare events, making it possible to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide insight into reweighting techniques.