Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation

A new simulation approach, called`subset simulation', is proposed to compute small failure probabilities encountered in reliability analysis of engineering systems. The basic idea is to express the failure probability as a product of larger conditional failure probabilities by introducing intermediate failure events. With a proper choice of the conditional events, the conditional failure probabilities can be made suf®ciently large so that they can be estimated by means of simulation with a small number of samples. The original problem of calculating a small failure probability, which is computationally demanding, is reduced to calculating a sequence of conditional probabilities, which can be readily and ef®ciently estimated by means of simulation. The conditional probabilities cannot be estimated ef®ciently by a standard Monte Carlo procedure, however, and so a Markov chain Monte Carlo simulation (MCS) technique based on the Metropolis algorithm is presented for their estimation. The proposed method is robust to the number of uncertain parameters and ef®cient in computing small probabilities. The ef®ciency of the method is demonstrated by calculating the ®rst-excursion probabilities for a linear oscillator subjected to white noise excitation and for a ®ve-story nonlinear hysteretic shear building under uncertain seismic excitation. q 2001 Elsevier Science Ltd. All rights reserved.