Adaptive Importance Sampling Based on Fault Tree Analysis for Piecewise Deterministic Markov Process

Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling method based on a cross-entropy procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. They are well adapted to high-dimensional industrial systems. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry.