Langevin importance sampling for reliability analysis

Abstract

Importance Sampling (IS) is a widely used method for reliability analysis, designed to increase the frequency of samples from the failure domain by introducing a biased sampling density, known as the IS density. Recent Markov Chain simulation methods, such as Hamiltonian Monte Carlo (HMC), use artificial dynamics to improve sampling efficiency over conventional random-walk algorithms like Metropolis-Hastings. However, HMC can be inefficient in high-dimensional problems or when model evaluations are costly. This paper develops a novel approach, named Langevin IS, which reframes the inference problem in HMC as an optimization task for constructing the IS density. Central to this approach is the Langevin equation, which unifies various HMC variants within a general stochastic dynamics formulation. The proposed approach leverages Langevin dynamics to design a parametric IS density that approximately satisfies the associated Fokker–Planck equation. From this equation, a new distance measure is derived that incorporates geometric information absent in conventional criteria like the Kullback–Leibler divergence. An efficient algorithm is developed to solve the resulting optimization problem, incorporating surrogate modeling and active learning to reduce computational cost. A theoretical guarantee is also provided, showing that the estimation error is bounded in terms of the surrogate approximation error. The effectiveness of Langevin IS is demonstrated through benchmark reliability problems, highlighting its ability to deliver accurate failure probability estimates with improved efficiency.