Rare event estimation of high dimensional problems with confidence intervals

Rare events have low probabilities of occurrence, and computation of such small probabilities is challenging, especially for high dimensional problems. Meanwhile, the robust estimation of the probability with narrow bounds is a key component for rare event estimation. Thus, confidence intervals of the estimator can be established based on the central limit theorem. Yet, the commonly used Monte Carlo simulation method is computationally inefficient as the sample size would be huge to derive a reasonably narrow confidence interval. Therefore, this paper introduces an efficient probability estimation technique that estimate the probability of rare events for high dimensional systems with smaller sample size and provide narrow estimation confidence intervals simultaneously. The asymptotic normality for the estimator has been proved theoretically without strong assumptions, and based on that, an asymptotic confidence interval has been established for the proposed probability estimator. The efficiency of the developed technique for probability estimation with confidence intervals is assessed with several engineering reliability analysis and design examples. Our numerical experiments demonstrate that a narrow confidence interval can be built efficiently with the probability estimation, and the real probability results always located within the proposed estimation bounds, which improve both efficiency and accuracy of the rare events estimation.