Efficient Markov Chain Monte Carlo for combined Subset Simulation and nonlinear finite element analysis

Typical probabilistic problems in an engineering context include rare event probability estimation for physical models where spatial autocorrelation of material property parameters is significant. Subset Simulation, a Markov Chain Monte Carlo technique, can be used to estimate rare event probabilities in physical models more efficiently than Monte Carlo Simulation. This efficiency gain is important when the sampling operation is computationally demanding, as is the case in the solution of stochastic Partial Differential Equations. In high dimensional spaces where Polynomial Chaos or other direct integration techniques become intractable, sampling methods may be the only way to compute integral functions in probabilistic analysis. In this paper, Subset Simulation is applied to probability of failure estimation in nonlinear elasto-plastic finite element problems. Further, a derivation of confidence intervals for Subset Simulation relative errors is presented. This new technique allows for vastly improved efficiency in the computation of error estimates for Subset Simulation. Significantly, the numerical studies presented indicate that for the tested finite element problems, Metropolis-Hastings sampling can outperform Componentwise Metropolis-Hastings and Gibbs sampling. This result is relevant to the design of efficient Subset Simulation methodologies.