Estimation of small failure probability using generalized subset simulation

This paper proposes a generalized subset simulation (GSS) method for estimating small failure probability. The basic idea is to modify the failure threshold and amplify the variability of input variables simultaneously in a sequential way. With a proper choice of the intermediate failure thresholds and amplification factors, the problem of estimation of a small failure probability is decomposed into the problem of estimating a series of simple integrals. For high-dimensional problems, two coordinate rotation schemes are introduced to detect the important directions in high-dimensional space. Using coordinate rotation, original high-dimensional integrals are transformed into a series of low-dimensional ones, and they can be estimated by Markov Chain Monte Carlo (MCMC) simulation technique efficiently. Two MCMC algorithms are introduced within GSS for low-dimensional and high-dimensional problems respectively, and a resampling method is utilized to select the initial states of each Markov chain from the existing samples. Five widely used benchmark examples are used to test the performance of the proposed method.