An adaptive optimal importance sampling method for efficiently calibrating augmented failure probability

Augmented failure probability (AFP) can measure the safety degree in case of random inputs with random distribution parameters, and solving the AFP updating model on gradually newly available observations can obtain posterior AFP to calibrate safety degree. However, there lack efficient algorithms for solving posterior AFP at present. Thus, an importance sampling (IS) method is proposed for efficiently estimating the posterior AFP when the newly available observations are gradually collected. By minimizing the variance of posterior AFP estimation, the optimal IS density is derived in the proposed IS method. For the operating difficulty resulted from the implicit character of the optimal IS density, the surrogate model of performance function is established for constructing a quasi-optimal IS density to adaptively approach the optimal one. For the sampling difficulty resulted from the irregularity of the quasi-optimal IS density, a new acceptance-rejection strategy is creatively designed, and its correctness is proved analytically. Since the proposed IS method adaptively combines the economic surrogate model with the minimizing estimation variance technique, the efficiency is greatly improved for estimating the posterior AFP. The examples show that the quantitative results of the posterior AFP are consistent with the qualitative ones. And comparing with the existing methods, the proposed IS method can improve the efficiency of sequentially estimating the posterior AFP while ensuring accuracy.