  This work introduces a novel framework for precisely and efficiently
estimating rare event probabilities in complex, high-dimensional non-Gaussian
spaces, building on our foundational Approximate Sampling Target with
Post-processing Adjustment (ASTPA) approach. An unnormalized sampling target is
first constructed and sampled, relaxing the optimal importance sampling
distribution and appropriately designed for non-Gaussian spaces. Post-sampling,
its normalizing constant is estimated using a stable inverse importance
sampling procedure, employing an importance sampling density based on the
already available samples. The sought probability is then computed based on the
estimates evaluated in these two stages. The proposed estimator is
theoretically analyzed, proving its unbiasedness and deriving its analytical
coefficient of variation. To sample the constructed target, we resort to our
developed Quasi-Newton mass preconditioned Hamiltonian MCMC (QNp-HMCMC) and we
prove that it converges to the correct stationary target distribution. To avoid
the challenging task of tuning the trajectory length in complex spaces,
QNp-HMCMC is effectively utilized in this work with a single-step integration.
We thus show the equivalence of QNp-HMCMC with single-step implementation to a
unique and efficient preconditioned Metropolis-adjusted Langevin algorithm
(MALA). An optimization approach is also leveraged to initiate QNp-HMCMC
effectively, and the implementation of the developed framework in bounded
spaces is eventually discussed. A series of diverse problems involving high
dimensionality (several hundred inputs), strong nonlinearity, and
non-Gaussianity is presented, showcasing the capabilities and efficiency of the
suggested framework and demonstrating its advantages compared to relevant
state-of-the-art sampling methods.
