Application of MCMC samplers with replica exchange and Russian Roulette in estimating structural reliability via importance sampling.

Abstract

Rapid changes in performance function values near regions of failure, the presence of multiple failure regions of importance, and several disconnected failure regions separated by large regions of safety can pose substantial difficulties in the accurate estimation of structural reliability. Within Monte Carlo simulation-based methods, even widely studied importance sampling (IS) schemes can fail to characterize the actual failure regions leading to poor estimates of failure probability. This study addresses two issues. Firstly, it proposes an improved ISpdf that utilizes a variant of Markov chain Monte Carlo (MCMC) sampler based on replica-exchange. In this method, in addition to the chain that draws from the target pdf, a secondary chain is run in parallel, which assists the original chain in an enhanced exploration of the state space. This improves the chances of detection of all significant failure regions even when the above-mentioned difficulties are present. The proposed IS scheme closely follows the algorithm developed by Au and Beck (1999) where the ISpdf is taken as the kernel density estimate constructed using samples generated by running an MCMC sampler with target pdf as the ideal ISpdf. The current study proposes that the more sophisticated replica-exchange based MCMC algorithm be used for generation of these samples. Secondly, the study builds upon this IS scheme and further reduces sampling variance using the Russian Roulette (RR) procedure, which seems to have remained unexplored in the context of reliability integral estimation. Herein, the IS estimate of failure probability is viewed as a weighted sum of indicator functions. It leverages the fact that if the weights of all samples are approximately equal, sampling variance is reduced. Thus, it is desirable to probabilistically kill/annihilate samples that contribute insignificantly to the sum, thus avoiding calling the performance function. A FORM-based strategy is used in the proper selection of the killing probability. The proposed IS scheme is applied to reliability estimation to a suite of problems drawn from the existing literature in which performance functions contain the mentioned difficulties. It is demonstrated that such difficulties in performance functions arise in problems of structural vibration and where failure is identified by instability or buckling. The proposed method is shown to identify all important failure regions and produce accurate estimates of failure probability. The results are compared with existing state-of-the-art IS schemes which are not guaranteed to provide acceptable estimates of failure probability when such geometrically complicated performance functions are considered.