Hamiltonian MCMC methods for estimating rare events probabilities in high-dimensional problems

TL;DR

This paper introduces a novel framework combining ASTPA with gradient-based Hamiltonian MCMC methods, including a new QNp-HMCMC scheme, to efficiently estimate rare event probabilities across various dimensions, outperforming existing methods.

Contribution

The paper develops a new QNp-HMCMC algorithm integrated within the ASTPA framework for improved rare event probability estimation in high-dimensional spaces.

Findings

QNp-HMCMC outperforms traditional MCMC methods in high-dimensional problems.

The proposed approach accurately estimates rare event probabilities with fewer samples.

Performance comparisons show advantages over Subset Simulation in challenging scenarios.

Abstract

Accurate and efficient estimation of rare events probabilities is of significant importance, since often the occurrences of such events have widespread impacts. The focus in this work is on precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on an introduced framework termed Approximate Sampling Target with Post-processing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. The developed techniques in this paper are applicable from low- to high-dimensional stochastic spaces, and the basic idea is to construct a relevant target distribution by weighting the original random variable space through a one-dimensional output likelihood model, using the limit-state function. To sample from this target distribution, we exploit HMCMC algorithms, a family of MCMC methods that adopts physical system dynamics, rather than solely using a proposal probability distribution, to generate distant sequential samples, and we develop a new Quasi-Newton mass preconditioned HMCMC scheme (QNp-HMCMC), which is particularly efficient and suitable for high-dimensional spaces. To eventually compute the rare event probability, an original post-sampling step is devised using an inverse importance sampling procedure based on the already obtained samples. The statistical properties of the estimator are analyzed as well, and the performance of the proposed methodology is examined in detail and compared against Subset Simulation in a series of challenging low- and high-dimensional problems.