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 sufficiently 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 efficiently estimated by means of simulation. The conditional probabilities cannot be estimated efficiently 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 efficient in computing small probabilities. The efficiency of the method is demonstrated by calculating the first-excursion probabilities for a linear oscillator subjected to white noise excitation and for a five-story nonlinear hysteretic shear building under uncertain seismic excitation.