  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.
