Monte Carlo Statistical Methods (Springer Texts in Statistics)
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Monte Carlo Strategies in Scientific Computing
Monte Carlo Strategies in Scientific Computing
Reliability-based design sensitivity by efficient simulation
Computers and Structures
On the use of a class of interior point algorithms in stochastic structural optimization
Computers and Structures
Global optimization using the asymptotically independent Markov sampling method
Computers and Structures
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Estimation of small failure probabilities is one of the most important and challenging computational problems in reliability engineering. The failure probability is usually given by an integral over a high-dimensional uncertain parameter space that is difficult to evaluate numerically. This paper focuses on enhancements to Subset Simulation (SS), proposed by Au and Beck, which provides an efficient algorithm based on MCMC (Markov chain Monte Carlo) simulation for computing small failure probabilities for general high-dimensional reliability problems. First, we analyze the Modified Metropolis algorithm (MMA), an MCMC technique, which is used in SS for sampling from high-dimensional conditional distributions. The efficiency and accuracy of SS directly depends on the ergodic properties of the Markov chains generated by MMA, which control how fast the chain explores the parameter space. We present some observations on the optimal scaling of MMA for efficient exploration, and develop an optimal scaling strategy for this algorithm when it is employed within SS. Next, we provide a theoretical basis for the optimal value of the conditional failure probability p"0, an important parameter one has to choose when using SS. We demonstrate that choosing any p"0@?[0.1,0.3] will give similar efficiency as the optimal value of p"0. Finally, a Bayesian post-processor SS+ for the original SS method is developed where the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples from SS are viewed as informative data relevant to the system's reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the information in the sampled data. This PDF quantifies the uncertainty in the value of the failure probability and it may be further used in risk analyses to incorporate this uncertainty. To demonstrate SS+, we consider its application to two different reliability problems: a linear reliability problem and reliability analysis of an elasto-plastic structure subjected to strong seismic ground motion. The relationship between the original SS and SS+ is also discussed.