Hamiltonian MCMC Based Framework for Time-variant Rare Event Uncertainty Quantification

Abstract

Accurate and efficient rare event uncertainty quantification is of ever-increasing significance, including for natural hazard cases characterized by substantial dynamic loads. In this work, our computationally efficient framework, termed Approximate Sampling Target with Post-processing Adjustment (ASTPA) is further examined, having exceptional performance in general static problems, including challenging cases of high-dimensionality, multi-modality, non-Gaussianity, and very small probabilities of failure. As shown in this work, ASTPA maintains its effectiveness and applicability even on complex first-passage dynamic problems. In their general form, these problems can usually be high-dimensional, due to the system uncertainties and the random variables involved in simulating the dynamic excitations as stochastic processes. In the suggested ASTPA framework, the multi-dimensional random variable space is weighted by a cumulative distribution function that utilizes the limit-state expression to construct an approximate sampling target distribution, providing increased importance to the failure domain. While any appropriate sampling scheme can be used to sample this constructed target distribution, Hamiltonian Markov Chain Monte Carlo (HMCMC) samplers have shown incomparable efficiency in this task, particularly our developed Quasi-Newton mass preconditioned Hamiltonian MCMC (QNp-HMCMC) approach. This efficient HMCMC variant is mainly based on constructing a suitable mass matrix, describing the local structure of the target distribution, by gradually forming in the adaptive burn-in stage of the algorithm a Hessian matrix based on Quasi-Newton principles. This Hessian information is also used to effectively guide the sampler towards promising regions of the domain. Given that the acquired samples are drawn from an approximate target, a post-processing adjustment is performed through a devised original inverse importance sampling (IIS) procedure, utilizing an importance sampling density appropriately constructed based on the already acquired HMCMC samples. An approximate analytical expression for the uncertainty of the computed estimator can thus be also derived based on importance sampling practices. In this work, the ASTPA framework is carefully examined on first-passage dynamic problems, and is successfully applied directly on both Gaussian and non-Gaussian stochastic spaces, an advantageous feature when the transformation to the favorable and preferred Gaussian space is unachievable. The performance of the discussed approach is tested and compared against the state-of-art Subset Simulation method in a series of high-dimensional, non-linear, stochastic dynamic problems.