Instanton based importance sampling for rare events in stochastic PDEs

Abstract

We present a new method for sampling rare and large fluctuations in a non-equilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.

arXiv comment8 pages, 4 figures

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• Collapse of transitional wall turbulence captured using a rare events algorithm
• Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems
• A Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation
• Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Synthetic Turbulence via an Instanton Gas Approximation