Relaxed subset simulation for reliability estimation

The practical implementation of subset simulation (SuS) may be biased in estimating the small failure probability, when the limit state function (LSF) exhibits pathological geometries that hinder the ergodicity of Markov Chain Monte Carlo (MCMC) sampling. To address this, we propose a “relaxed†version of SuS (Re-SuS), which replaces the conventional indicator function with a hybrid indicator function to enhance the capability of MCMC in exploring the standard normal space. This modification stems from a sequential importance sampling (SIS) interpretation of SuS, offering flexibility in selecting intermediate sampling distributions (ISDs). The hybrid indicator function incorporates both the LSF and the probability density function (PDF) of standard normal variables, acknowledging that failure events typically correspond to small PDF values. By including portions of intermediate safe domains, the ISD in Re-SuS slows the transition to the failure domain, improving the likelihood of identifying true design points. Various benchmark examples are presented to validate the performance of Re-SuS, demonstrating its robustness against misleading LSF geometries. While the estimation uncertainty is marginally higher than in the original SuS, Re-SuS significantly reduces the potential bias in failure probability estimation. More broadly, the SIS interpretation of SuS provides opportunities for further performance enhancements through careful design of the ISD.