Hierarchical importance sampling method for estimating failure probability function

Failure probability function can quantify the structural safety levels when the random input distribution parameter varies within the concerned design region, and it can decouple the reliability-based design optimization model. To efficiently estimate it, a hierarchical importance sampling method is proposed. The main contributions of this paper are threefold. The first is constructing a novel single-loop optimal importance sampling density, on which the double-loop framework of analyzing the failure probability function is decoupled. The second is employing Markov Chain Monte Carlo simulation to extract the samples of the single-loop optimal importance sampling density in an adaptively hierarchical way. Compared to the existing single-loop cross-entropy based importance sampling method, the proposed method eliminates iterative determination of the unknown parameter set of the Gaussian mixture model. The third is utilizing a Gaussian mixture model to inversely approximate the single-loop optimal importance sampling density, enabling efficient and accurate estimation of the failure probability function. By integrating the single-loop strategy, importance sampling variance reduction technique, and inverse approximation of Gaussian mixture model, the proposed method improves the efficiency of estimating failure probability function while maintaining the accuracy, which is rigorously validated through five examples, demonstrating the superior performance compared to state-of-the-art alternatives.