Reliability Estimation for Dynamical Systems Subject to Stochastic Excitation using Subset Simulation with Splitting

Abstract

A new Subset Simulation approach is proposed in this paper for reliability estimation for dynamical systems subject to stochastic excitation. The basic idea of Subset Simulation is to consider a small failure probability as a product of larger failure probabilities conditional on intermediate failure events. This new approach does not require Markov Chain Monte Carlo simulation, in contrast to the original method, to generate conditional samples for estimating the conditional probabilities; instead, only direct Monte Carlo simulation is needed. The method employs splitting of a trajectory that reaches an intermediate failure level into multiple trajectories subsequent to its first passage time. This exploits an important feature of causal dynamical systems, namely, the distribution of the future excitation subsequent to the first passage time and conditional on the previous excitation is just equal to its unconditional counterpart. The statistical properties of the failure probability estimates are presented, where it is shown that the estimates are unbiased and formulas are derived to assess the error of estimation, including the coefficient of variation of the estimates. The resulting algorithm is simple and easy to implement. Two examples are presented to demonstrate the effectiveness of the new approach, also to compare with the original Subset Simulation and with direct Monte Carlo simulation.