Transitional active learning of small probabilities

Efficient estimation of small failure probability subjected to multiple failure domains is one of the central and challenging issues in structural reliability analysis and other rare event analysis tasks, especially in case where the computational resource is quite limited but high accuracy is required. A new active learning scheme, named as Transitional Bayesian Quadrature (TBQ), is proposed to fill this gap. Leveraging two types of smooth Artificial Intermediate Distributions (AIDs) for sequentially approaching the optimal importance sampling density, a Bayesian quadrature technique equipped with two novel acquisition functions is proposed for adaptive specification of the tempering parameters of the AIDs and active learning of the ratios of successive intermediate probabilities, with desired accuracy. Of special contribution is the presentation of closed-form formulations for facilitating the numerical computations concerning both acquisition functions and quadrature rules, making the TBQ algorithms numerically efficient and robust. A bridging scheme is also introduced for improving the stability. Two benchmark studies and two engineering applications are ultimately presented for demonstrating the effectiveness and relative merits of the two TBQ algorithms.