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

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 multi-scale 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 order-of-magnitudes, the probability to sample extreme and rare events, making it possible for the first time 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 a new insight on reweighting techniques.

A Unifying Framework for Describing Rogue Waves

“…Because of this key property, the instanton is the natural object for the characterization of the extreme wave events. Note that the knowledge of the instanton configuration itself can be used as an ingredient for advanced rare event sampling techniques, such as importance sampling and hybrid Monte Carlo approaches [50]. For the purpose of this paper, we restrict our analysis to the comparison of the instanton to the conditioned experimental measurements.…”

Instanton based importance sampling for rare events in stochastic PDEs

“…A subsequent reweighting procedure allows us to calculate the full PDF with a much more accurate description of the tails in comparison to ordinary direct numerical simulations (DNS) of Burgers turbulence. We also show that our method is computationally less substantially challenging than other approaches based on Markov Chain Monte Carlo methods to generate extreme and rare flow configurations 57 . Hence, the method can be considered as an optimal application of rare events importance sampling and we call it the Instanton based Importance Sampling (IbIS), see also the work 58 for a similar idea.…”

“…Section III constitutes the core of the paper, where we present our reweighting procedure. In section IV we describe in detail the numerical protocol and we compare the results obtained with IbIS against those obtained using DNS and a Hybrid Monte Carlo approach 57 . We close with a summary and an outlook on further applications.…”

Zeta-regularized vacuum expectation values fromquantum computing simulations

“…of important observables Ω in high energy physics, statistical mechanics and beyond, e.g. in turbulence [3]. In eq.…”