A novel double-loop variance reduction technique for estimating reliability based on dimensionality reduction and cross-entropy-based importance sampling

To enhance the efficiency of single-loop variance reduction methods in structural reliability analysis, we propose a novel double-loop variance reduction method that integrates dimensionality reduction with cross-entropy-based importance sampling. In the first loop, the variance reduction is realized by transforming the failure probability into the expectation of the conditional failure probability, which can be analytically solved by the cumulative distribution function of one-dimensional reduction input, with respect to the remaining input vector by removing the one-dimensional reduction input. In the second loop, the variance reduction is realized by approaching the theoretically optimal importance sampling density for estimating the expectation transformed in the first loop, and a Gaussian mixture model is employed to approach this optimal density, where the parameters of Gaussian mixture model are optimized by minimizing the Kullback-Leibler cross-entropy between Gaussian mixture model and the theoretically optimal importance sampling density. Additionally, Kriging surrogate model of the performance function is embedded within the proposed double-loop architecture to decrease the number of costly performance function evaluations, significantly enhancing computational efficiency. The principal innovation of this study lies in the integration of dimensionality reduction with cross-entropy-based importance sampling within a double-loop strategy, providing a robust and efficient strategy for failure probability estimation.