Rare event simulation via importance sampling for linear SPDE's

TL;DR

This paper develops efficient importance sampling methods for estimating rare events in linear SPDEs, leveraging spectral gaps to reduce dimensionality and improve performance in simulations.

Contribution

It introduces a provably effective importance sampling approach for linear SPDEs that remains robust in infinite-dimensional settings and for small noise levels.

Findings

Importance sampling performs well even in infinite dimensions.

Spectral gap conditions enable dimensionality reduction.

Simulation results confirm theoretical efficiency.

Abstract

The goal of this paper is to develop provably efficient importance sampling Monte Carlo methods for the estimation of rare events within the class of linear stochastic partial differential equations (SPDEs). We find that if a spectral gap of appropriate size exists, then one can identify a lower dimensional manifold where the rare event takes place. This allows one to build importance sampling changes of measures that perform provably well even pre-asymptotically (i.e. for small but non-zero size of the noise) without degrading in performance due to infinite dimensionality or due to long simulation time horizons. Simulation studies supplement and illustrate the theoretical results.