Large deviation theory-based adaptive importance sampling for rare events in high dimensions

TL;DR

This paper introduces a novel high-dimensional rare event probability estimation method combining large deviation theory with adaptive importance sampling, improving efficiency without tuning parameters.

Contribution

It develops a large deviation theory-based adaptive importance sampling approach that identifies informative subspaces and reuses samples for accurate rare event estimation.

Findings

Outperforms existing importance sampling schemes in high dimensions.

Effectively identifies low-dimensional subspaces for rare event analysis.

Demonstrates applicability to complex physical models.

Abstract

We propose a method for the accurate estimation of rare event or failure probabilities for expensive-to-evaluate numerical models in high dimensions. The proposed approach combines ideas from large deviation theory and adaptive importance sampling. The importance sampler uses a cross-entropy method to find an optimal Gaussian biasing distribution, and reuses all samples made throughout the process for both, the target probability estimation and for updating the biasing distributions. Large deviation theory is used to find a good initial biasing distribution through the solution of an optimization problem. Additionally, it is used to identify a low-dimensional subspace that is most informative of the rare event probability. This subspace is used for the cross-entropy method, which is known to lose efficiency in higher dimensions. The proposed method does not require smoothing of indicator functions nor does it involve numerical tuning parameters. We compare the method with a state-of-the-art cross-entropy-based importance sampling scheme using three examples: a high-dimensional failure probability estimation benchmark, a problem governed by a diffusion equation, and a tsunami problem governed by the time-dependent shallow water system in one spatial dimension.