Sequential Sampling-Based Asymptotic Probability Estimation of High-Dimensional Rare Events

Abstract

Accurate analysis of rare failure events with an affordable computational cost is often challenging in many engineering applications, particularly for problems with high-dimensional system inputs. The extremely low probabilities of occurrence often lead to large probability estimation errors and low computational efficiency. Thus, it is vital to develop advanced probability analysis methods that are capable of providing robust estimations of rare event probabilities with narrow confidence bounds. The general method of determining confidence intervals of an estimator using the central limit theorem faces the critical obstacle of low computational efficiency. This is a side effect of the widely used Monte Carlo method, which often requires a large number of simulation samples to derive a reasonably narrow confidence interval. In this paper, a new probability analysis approach is developed which can be used to derive the estimates of rare event probabilities efficiently with narrow estimation bounds simultaneously for high-dimensional problems and complex engineering systems. The asymptotic behavior of the developed estimator is proven theoretically without imposing strong assumptions. An asymptotic confidence interval is established for the developed estimator. The presented study offers important insights into the robust estimations of the probability of occurrences for rare events. The accuracy and computational efficiency of the developed technique are assessed with numerical and engineering case studies. Case study results have demonstrated that narrow bounds can be obtained efficiently using the developed approach with the true values consistently located within the estimation bounds.