Selecting Reduced Models in the Cross-Entropy Method

TL;DR

This paper introduces a method to efficiently estimate rare event probabilities using importance sampling with the cross-entropy method, by adaptively selecting low-dimensional approximations of computationally expensive score functions based on error bounds.

Contribution

It proposes a hierarchical approach to select the simplest yet sufficiently accurate score function approximation during the CE optimization, reducing computational costs.

Findings

Significant reduction in computation time demonstrated.

Effective in PDE-based pollution alert systems.

Theoretical guarantees on convergence and efficiency.

Abstract

This paper deals with the estimation of rare event probabilities using importance sampling (IS), where an optimal proposal distribution is computed with the cross-entropy (CE) method. Although, IS optimized with the CE method leads to an efficient reduction of the estimator variance, this approach remains unaffordable for problems where the repeated evaluation of the score function represents a too intensive computational effort. This is often the case for score functions related to the solution of a partial differential equation (PDE) with random inputs. This work proposes to alleviate computation by the parsimonious use of a hierarchy of score function approximations in the CE optimization process. The score function approximation is obtained by selecting the surrogate of lowest dimensionality, whose accuracy guarantees to pass the current CE optimization stage. The selection relies on certified upper bounds on the error norm. An asymptotic analysis provides some theoretical guarantees on the efficiency and convergence of the proposed algorithm. Numerical results demonstrate the gain brought by the method in the context of pollution alerts and a system modeled by a PDE.