Estimation of small failure probabilities by partially Bayesian active learning line sampling: Theory and algorithm

Line sampling (LS) has proved to be a highly promising advanced simulation technique for assessing small failure probabilities. Despite the great interest in practical engineering applications, many efforts from the research community have been devoted to improving the standard LS. This paper aims at offering some new insights into the LS method, leading to an innovative method, termed ‘partially Bayesian active learning line sampling’ (PBAL-LS). The problem of evaluating the failure probability integral in the LS method is treated as a Bayesian, rather than frequentist, inference problem, which allows to incorporate our prior knowledge and model the discretization error. The Gaussian process model is used as the prior distribution for the distance function, and the posterior mean, and an upper bound of the posterior variance of the failure probability are derived. Based on the posterior statistics of the failure probability, we also put forward a learning function and a stopping criterion, which enable us to use active learning. Besides, an efficient algorithm is also designed to implement the PBAL-LS method, with the ability to automatically adjust the important direction and efficiently process the lines. Five numerical examples are studied to demonstrate the performance of the proposed PBAL-LS method against several existing methods.