A direct importance sampling-based framework for rare event uncertainty quantification in non-Gaussian spaces

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 random variables), 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.