Cross-entropy-based adaptive importance sampling using Gaussian mixture

Abstract

Structural reliability analysis frequently requires the use of sampling-based methods, particularly for the situation where the failure domain in the random variable space is complex. One of the most efficient and widely utilized methods to use in such a situation is importance sampling. Recently, an adaptive importance sampling method was proposed to find a near-optimal importance sampling density by minimizing Kullback–Leibler cross entropy, i.e. a measure of the difference between the absolute best sampling density and the one being used for the importance sampling. In this paper, the adaptive importance sampling approach is further developed by incorporating a nonparametric multimodal probability density function model called the Gaussian mixture as the importance sampling density. This model is used to fit the complex shape of the absolute best sampling density functions including those with multiple important regions. An efficient procedure is developed to update the Gaussian mixture model toward a near-optimal density using a small size of pre-samples. The proposed method needs only a few steps to achieve a near-optimal sampling density, and shows significant improvement in efficiency and accuracy for a variety of component and system reliability problems. The method requires far less samples than both crude Monte Carlo simulation and the cross-entropy-based adaptive importance sampling method employing a unimodal density function; thus achieving relatively small values of the coefficient of variation efficiently. The computational efficiency and accuracy of the proposed method are not hampered by the probability level, dimension of random variable space, and curvatures of limit-state function. Moreover, the distribution model parameters of the Gaussian densities in the obtained near-optimal density help identify important areas in the random variable space and their relative importance.