Langevin Dynamics for Importance Sampling in Reliability Analysis
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Armin Tabandeh, Gaofeng Jia, Paolo Gardoni, Langevin Dynamics for Importance Sampling in Reliability Analysis, 14th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP14), Dublin, Ireland, 2023.Download Item:
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.
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