Reliability estimation for dynamical systems subject to stochastic excitation using subset simulation with splitting

A new subset simulation approach is proposed for reliability estimation for dynamical systems subject to stochastic excitation. The basic idea of subset simulation is to factor a small failure probability into a product of larger failure probabilities conditional on intermediate failure events. The new method proposed in this work does not require Markov Chain Monte Carlo simulation, in contrast to the original method, to estimate 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 the corresponding first passage time. The new approach still enjoys most of the advantages of the original subset simulation, e.g. it is applicable to general causal dynamical systems and it is robust with respect to the dimension of the uncertain input variables. The statistical properties of the failure probability estimators are presented, where it is shown that they are unbiased and formulas are derived to assess the error of estimation, including the coefficient of variation. We also discuss the selection of intermediate failure events and the number of samples for each failure level. The resulting algorithm is simple and easy to implement. Two examples are presented to demonstrate the effectiveness of the new approach, and the results are compared with the original subset simulation and with direct Monte Carlo simulation.