Langevin Dynamics for Importance Sampling in Reliability Analysis

Abstract

Importance Sampling (IS) is a popular simulation method for the reliability analysis of general engineering systems. Its core idea is to build a biased density that can sample from the failure domain more frequently than in the Monte Carlo simulation. The optimal IS density, derived from the Euler-Lagrange equation, is inversely proportional to the unknown failure probability to be estimated; thus, directly sampling from it is impossible. However, samples can indirectly be drawn from the optimal IS density by simulating a Markov chain. In particular, the Hamiltonian Monte Carlo (HMC) sampling algorithm and its variants design a dynamical system that can explore the failure domain better than, for example, the Metropolis-Hastings algorithm. However, such sampling algorithms can be prohibitively slow if the problem is high-dimensional or involves expensive model evaluations. This paper presents a method to convert the inference problem in HMC into an optimization problem and discusses its connection with existing optimization-based algorithms for IS. The starting point is the Langevin equation, which offers a unified formulation for several variants of HMC. The optimal IS density is the steady-state solution of the Fokker-Planck equation associated with the Langevin equation. The proposed method approximates the steady-state solution of the Fokker-Planck equation via optimization. The process is illustrated through a benchmark reliability problem.