Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions

• Published in: Computers & structures Vol. 92; pp. 283 - 296
• Main Authors: Zuev, Konstantin M., Beck, James L., Au, Siu-Kui, Katafygiotis, Lambros S.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.02.2012; Elsevier
• Subjects: Algorithms; Applied sciences; Bayesian approach; Computer simulation; Estimating; Exact sciences and technology; Failure; Fluctuation phenomena, random processes, noise, and brownian motion; Markov chain Monte Carlo; Mathematical models; Mathematics; Monte Carlo methods; Operational research and scientific management; Operational research. Management science; Optimization; Physics; Polymethyl methacrylates; Probability and statistics; Probability theory and stochastic processes; Reliability engineering; Reliability theory. Replacement problems; Risk theory. Actuarial science; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistical physics, thermodynamics, and nonlinear dynamical systems; Stochastic simulation methods; Subset Simulation; Reliability engineering; Stochastic simulation methods; Subset Simulation; Bayesian approach; Markov chain Monte Carlo; Ergodic theory; Stochastic model; Conditional distribution; Conditional probability; Stochastic method; Modeling; Parameter space; Markov chain; System with n degrees of freedom; Optimal strategy; Uncertain system; Elastoplasticity; Sampling; Bayes estimation; Metropolis algorithm; Prior information; Monte Carlo method; Imprecise probability; Rupture; Ergodicity; Risk analysis; System reliability
• DOI: 10.1016/j.compstruc.2011.10.017
• ISSN: 0045-7949

► The Modified Metropolis algorithm (MMA) for sampling from complex PDFs is analyzed. ► An optimal scaling strategy for MMA is presented. ► A theoretical basis for the optimal value of the conditional probability in Subset Simulation is provided. ► A Bayesian post-processor for the original Subset Simulation method is developed. ► The relationship between Subset Simulation and its Bayesian post-processor is given. 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.