Improved cross entropy-based importance sampling with a flexible mixture model

The probability of a rare event or failure event is defined through a potentially high-dimensional integral, whose integration domain is often only known point-wise in terms of the outcome of a numerical model. The probability of failure can be estimated efficiently through importance sampling (IS), provided that an effective IS density is chosen. The cross entropy (CE) method is an adaptive sampling approach that determines the IS density through minimizing the Kullback-Leibler divergence between the theoretically optimal IS density and a chosen parametric family of distributions. We propose an improved version of the classical CE method that introduces a smooth transition to make better use of the samples from intermediate sampling levels for fitting the sought IS density. The improved CE method is combined with a novel flexible parametric distribution model that is able to handle low- and high-dimensional problems as well as problems with multimodal failure domains. A set of numerical examples demonstrate that the proposed approach performs consistently better than the classical CE method in various problem settings.