Multilevel Sequential Importance Sampling for Rare Event Estimation

TL;DR

This paper introduces a novel, efficient Sequential Importance Sampling method for estimating rare event probabilities involving PDEs, overcoming computational challenges and nestedness issues of existing approaches.

Contribution

It develops a new adaptive Sequential Importance Sampling estimator that handles multiple discretization levels and introduces a novel MCMC kernel using von Mises-Fischer-Nakagami proposals.

Findings

The proposed method outperforms Multilevel Subset Simulation in test problems.

It effectively manages computational costs for PDE-based failure probability estimation.

The new MCMC kernel improves sampling efficiency in high-dimensional spaces.

Abstract

The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit state function, which depends on the solution of a partial differential equation (PDE). Since numerical evaluations of PDEs are computationally expensive, estimating such probabilities of failure by Monte Carlo sampling is intractable. More efficient sampling methods from reliability analysis, such as Subset Simulation, are popular, but can still be impracticable if the PDE evaluations are very costly. In this article, we develop a novel, highly efficient estimator for probabilities of rare events. Our method is based on a Sequential Importance sampler using discretizations of PDE-based limit state functions with different accuracies. A twofold adaptive algorithm ensures that we obtain an estimate based on the desired discretization accuracy. In contrast to the Multilevel Subset Simulation estimator of [Ullmann, Papaioannou 2015; SIAM/ASA J. Uncertain. Quantif. 3(1):922-953], our estimator overcomes the nestedness problem. Of particular importance in Sequential Importance sampling algorithms is the correct choice of the MCMC kernel. Instead of the popular adaptive conditional sampling method, we propose a new algorithm that uses independent proposals from an adaptively constructed von Mises-Fischer-Nakagami distribution. The proposed algorithm is applied to test problems in 1D and 2D space, respectively, and is compared to the Multilevel Subset Simulation estimator and to single-level versions of Sequential Importance Sampling and Subset Simulation.