Bayesian Multilevel Model Calibration for Inverse Problems Under Uncertainty with Perfect Data

A probabilistic framework for Bayesian inference and uncertainty analysis is developed. It allows inverse problems to be addressed in experimental situations where data are scarce and uncertainty is ubiquitous. The uncertainty characterization subproblemof the NASA Langley Multidisciplinary Uncertainty Quantification Challenge serves as the motivating application example. From the responses of a computational model, the goal is to learn about unknown model inputs that are subject to multiple types of uncertainty. This objective is interpreted and solved as Bayesian multilevel model calibration. The zero-noise or "perfect" data limit is investigated. Thereby, the likelihood function is defined as a solution to forward uncertainty propagation. Posterior explorations are based on suitable Markov chain Monte Carlo algorithms and stochastic likelihood simulations. An unforeseen finding in this context is that the posterior distribution can only be sampled with a certain degree of fidelity. Partial data augmentation is introduced as a means to improve the error statistics of likelihood estimations and the fidelity of posterior computations.