Extreme event quantification in dynamical systems with random components
Provides a principled framework for quantifying extreme event probabilities in uncertain dynamical systems, addressing a key challenge in uncertainty quantification.
The paper develops a method combining large deviation theory and optimal control to estimate the probability of extreme events in dynamical systems with random parameters or initial conditions, demonstrating efficiency on a rod with random elasticity and the nonlinear Schrödinger equation.
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 Schrödinger equation (NLSE) with random initial conditions.