Cross-entropy-based adaptive importance sampling using von Mises-Fisher mixture for high dimensional reliability analysis

• An adaptive importance sampling is developed using Kullback–Leibler cross entropy.
• The importance sampling density with minimum cross entropy is found by pre-samples.
• The von Mises-Fisher mixture model is utilized for applications to high dimensions.
• Simple formulas and rules are developed to facilitate minimizing the cross entropy.
• Superb performance is not affected by probability level, dimension or nonlinearity.

In order to address challenges in performing importance sampling in a high dimensional space of random variables, the paper develops a cross-entropy-based adaptive importance sampling technique that employs a von Mises-Fisher mixture as the sampling density model. By small-size pre-samplings, the proposed approach first finds a near-optimal sampling density by minimizing the Kullback–Leibler cross entropy between a von Mises-Fisher mixture model and the absolute best importance sampling density. To facilitate the minimization process, updating rules for parameters of the von Mises-Fisher mixture model are derived. Various practical issues associated with the updating rules are discussed and heuristic rules to improve the performance of the importance sampling are introduced. At the stage of final sampling, two slightly different sampling strategies are proposed to provide analysis options. Three numerical examples are investigated to test and demonstrate the proposed importance sampling method. The numerical examples show that the proposed approach, applicable to both component and system reliability problems, has superior performance for high dimensional reliability analysis problems with low failure probabilities.