A review and assessment of importance sampling methods for reliability analysis

This paper reviews the mathematical foundation of the importance sampling technique and discusses two general classes of methods to construct the importance sampling density (or probability measure) for reliability analysis. The paper first explains the failure probability estimator of the importance sampling technique, its statistical properties, and computational complexity. The optimal but not implementable importance sampling density, derived from the variational calculus, is the starting point of the two general classes of importance sampling methods. For time-variant reliability analysis, the optimal but not implementable stochastic control is derived that induces the corresponding optimal importance sampling probability measure. In the first class, the optimal importance sampling density is directly approximated by a member of a family of parametric or nonparametric probability density functions. This approximation requires defining the family of approximating probability densities, a measure of distance between two probability densities, and an optimization algorithm. In the second class, the approximating importance sampling density has the general functional form of the optimal solution. The approximation amounts to replacing the limit-state function with a computationally convenient surrogate. The paper then explores the performances of the two classes of importance sampling methods through several benchmark numerical examples. The challenges and future directions of the importance sampling technique are also discussed. • The mathematical foundation of the importance sampling (IS) technique is explained. • Failure probability estimators, their statistical properties, and computational complexity are presented. • Two classes of IS methods, called density and limit-state approximation methods, are explained. • The performances of the IS methods are explored through benchmark numerical examples. • The challenges and future directions of the IS methods are discussed.