Sequential and adaptive probabilistic integration for structural reliability analysis

• Sequential and adaptive probabilistic integration (SAPI) is proposed for structural reliability. • The devised SAPI estimator is an importance sampling (IS) estimator of the probability of failure. • Bayesian integration is performed on a sequence of distributions that gradually approaches the optimal IS density. • Posterior statistics of the normalising constant are quantified by random process sampling. • An active learning is designed to achieve a minimum number of limit state function evaluations. We propose the application of sequential and adaptive probabilistic integration (SAPI) to the estimation of the probability of failure in structural reliability. SAPI was originally developed to explore the posterior distribution and estimate its normalising constant in Bayesian model updating. The principle is to perform probabilistic integration on a sequence of distributions, moving from the prior to the posterior, to learn the normalising constant of each distribution. In structural reliability, SAPI can be used to sample an approximation of the optimal importance sampling (IS) density, and we present a particular choice of the intermediate distributions. The derived SAPI estimator is thus an IS estimator of the thought probability. The numerical uncertainty is propagated using random process sampling, and the induced posterior statistics are used to design a Bayesian active learning strategy. Four numerical examples demonstrate that SAPI outperforms other state-of-the-art active learning reliability methods using sequential Monte Carlo samplers.