Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

TL;DR

This paper introduces an importance sampling method for efficiently estimating exit probabilities of stochastic reaction-diffusion equations under moderate deviation scaling, combining linearization and finite-dimensional approximation.

Contribution

It develops a provably efficient importance sampling scheme leveraging linearization and finite-dimensional subspaces for reaction-diffusion equations in the moderate deviation regime.

Findings

Scheme performs well in zero noise limit

Effective in pre-asymptotic regimes

Validated through simulation studies

Abstract

We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.