  Estimating failure probabilities of engineering systems is an important
problem in many engineering fields. In this work we consider such problems
where the failure probability is extremely small (e.g $\leq10^{-10}$). In this
case, standard Monte Carlo methods are not feasible due to the extraordinarily
large number of samples required. To address these problems, we propose an
algorithm that combines the main ideas of two very powerful failure probability
estimation approaches: the subset simulation (SS) and the multicanonical Monte
Carlo (MMC) methods. Unlike the standard MMC which samples in the entire domain
of the input parameter in each iteration, the proposed subset MMC algorithm
adaptively performs MMC simulations in a subset of the state space and thus
improves the sampling efficiency. With numerical examples we demonstrate that
the proposed method is significantly more efficient than both of the SS and the
MMC methods. Moreover, the proposed algorithm can reconstruct the complete
distribution function of the parameter of interest and thus can provide more
information than just the failure probabilities of the systems.
