BIMC: The Bayesian Inverse Monte Carlo method for goal-oriented uncertainty quantification. Part II

TL;DR

This paper extends the Bayesian Inverse Monte Carlo method to handle highly nonlinear systems in rare-event probability estimation by introducing an adaptive, two-stage approach that uses optimization and Gaussian mixtures.

Contribution

The paper introduces the Adaptive-BIMC algorithm, enhancing BIMC's applicability to nonlinear systems through a two-stage process involving optimization and surrogate modeling.

Findings

A-BIMC effectively estimates rare-event probabilities in synthetic examples.

The method's limitations and failure conditions are systematically analyzed.

Use of local surrogates reduces computational costs.

Abstract

In Part I (arXiv:1911.00619) of this article, we proposed an importance sampling algorithm to compute rare-event probabilities in forward uncertainty quantification problems. The algorithm, which we termed the "Bayesian Inverse Monte Carlo (BIMC) method", was shown to be optimal for problems in which the input-output operator is nearly linear. But applying the original BIMC to highly nonlinear systems can lead to several different failure modes. In this paper, we modify the BIMC method to extend its applicability to a wider class of systems. The modified algorithm, which we call "Adaptive-BIMC (A-BIMC)", has two stages. In the first stage, we solve a sequence of optimization problems to roughly identify those regions of parameter space which trigger the rare-event. In the second stage, we use the stage one results to construct a mixture of Gaussians that can be then used in an importance sampling algorithm to estimate rare event probability. We propose using a local surrogate that minimizes costly forward solves. The effectiveness of A-BIMC is demonstrated via several synthetic examples. Yet again, the modified algorithm is prone to failure. We systematically identify conditions under which it fails to lead to an effective importance sampling distribution.