Cross entropy-based importance sampling for first-passage probability estimation of randomly excited linear structures with parameter uncertainty

Abstract

We propose an efficient importance sampling (IS) method for estimating the first-passage probability of linear structures with uncertain parameters and subjected to Gaussian process excitations. The method evaluates the reliability through integrating the conditional first-passage probability given the uncertain structural parameters. We develop an adaptive IS strategy to efficiently perform this integration based on an IS density that is constructed using the cross entropy (CE) method. The CE method determines the IS density by adaptively minimizing the Kullback–Leibler divergence between the theoretically optimal sampling density and a chosen parametric family of probability distributions. The CE optimization problem is solved for a series of target densities that gradually approach the optimal IS density of the structural parameters. To define the intermediate densities, a smoothening of the conditional first-passage probabilities is employed. Once the IS density of the uncertain structural parameters is obtained, an effective IS density of the random excitations conditional on the structural parameters is introduced to estimate the failure probability of the structure. Unlike other tailored methods for solving this problem, the proposed IS approach does not require any prior analysis of the dynamic system and can be applied as a black-box method. Numerical examples demonstrate that the proposed method can calculate the first-passage probability with remarkable efficiency.