T. Schorlepp, S. Tong, T. Grafke, and G. Stadler, Statistics and Computing **33** (2023), 137

#### Abstract

We introduce and compare computational techniques for sharp extreme
event probability estimates in stochastic differential equations
with small additive Gaussian noise. In particular, we focus on
strategies that are scalable, i.e. their efficiency does not degrade
upon spatial and temporal refinement. For that purpose, we extend
algorithms based on the Laplace method for estimating the
probability of an extreme event to infinite dimensions. The method
estimates the limiting exponential scaling using a single
realization of the random variable, the large deviation
minimizer. Finding this minimizer amounts to solving an optimization
problem governed by a differential equation. The probability
estimate becomes sharp when it additionally includes prefactor
information, which necessitates computing the determinant of a
second derivative operator to evaluate a Gaussian integral around
the minimizer. We present an approach in infinite dimensions based
on Fredholm determinants, and develop numerical algorithms to
compute these determinants efficiently for the high-dimensional
systems that arise upon discretization. We also give an
interpretation of this approach using Gaussian process covariances
and transition tubes. An example model problem, for which we also
provide an open-source python implementation, is used throughout the
paper to illustrate all methods discussed. To study the performance
of the methods, we consider examples of stochastic differential and
stochastic partial differential equations, including the randomly
forced incompressible three-dimensional Navier-Stokes equations.

doi:10.1007/s11222-023-10307-2

arXiv