Extreme event quantification in dynamical systems with random components

TL;DR

This paper develops a probabilistic framework combining large deviation theory and optimal control to predict the occurrence of extreme events in dynamical systems with uncertain parameters, demonstrated on physical models.

Contribution

It introduces a method to estimate and predict rare, extreme events in dynamical systems with random components using large deviation theory and optimal control techniques.

Findings

Extreme events can be characterized as minimizers of the LDT action functional.

Efficient numerical methods from optimal control can compute these minimizers.

Applications to physical models demonstrate the approach's effectiveness.

Abstract

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. Here this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system's parameters and/or its initial conditions. Specifically, it is established under which conditions such extreme events occur in a predictable way, as the minimizer of the LDT action functional. It is also shown how this minimization can be numerically performed in an efficient way using tools from optimal control. These findings are illustrated on the examples of a rod with random elasticity pulled by a time-dependent force, and the nonlinear Schr\"odinger equation (NLSE) with random initial conditions.