Adaptive proposal length scale in Subset Simulation

Subset Simulation (SS) is a Monte Carlo method for estimating the failure probability of a system whose response is a ‘black box’, for which little or no prior information is available for variance reduction. Pivotal to SS is an efficient mechanism for generating candidates that are accepted/rejected by Markov Chain Monte Carlo (MCMC) to produce an unbiased estimate. In the standard Normal space, conditional sampling scheme offers an elegant means for generating candidates, reducing the choice of proposal distribution in MCMC to a correlation parameter. Recent developments feature adaptive schemes to achieve some target acceptance rate. For a generic 1-D linear problem, this work obtains analytically the optimal correlation parameter that minimises the lag-1 correlation of samples in a simulation level of SS. Despite the 1-D linear origin, numerical investigations reveal that the resulting adaptive scheme shows promise for effectively suppressing the systematic growth of candidate rejection and correlation along Markov chains for problems of wider context, e.g., with nonlinearity, high dimensions and multiple failure modes. The adaptive scheme exhibits robustness for coping with complex problems where it is difficult to generate failure samples, although efficiency gain in variance reduction may be offset by increased correlation suspectedly between simulation levels. The analytical results derived in this work provide insights on how proposal PDFs should be scaled to cope with rare events.