Tim Wildey (Sandia National Lab, Albuquerque, NM) - A Consistent Bayesian Approach for Stochastic Inverse Problems

Uncertainty is ubiquitous in computational science and engineering. Often, parameters of interest cannot be measured directly and must be inferred from observable data. The mapping between these parameters and the measureable data is often referred to as the forward model and the goal is to use the forward model to gain knowledge about the parameters given the observations on the data. Statistical Bayesian inference is the most common approach for incorporating stochastic data into probabilistic descriptions of the input parameters. This particular approach uses data and an assumed error model to inform posterior distributions of model inputs and model discrepancies. An explicit characterization of the posterior distribution is not necessary since certain sampling methods, such as Markov Chain Monte Carlo, can be used to draw samples from the posterior. We have recently developed an alternative Bayesian solution to the stochastic inverse problem. We use measure-theoretic principles to prove that this approach produces a posterior distribution that is consistent in the sense that the push-forward probability density of the posterior through the model will match the distribution on the observable data, i.e., the posterior is consistent with the model and the data. The approach requires approximating the push-forward probability density of the prior through the computational model, which is fundamentally a forward propagation of uncertainty. We employ advanced approaches for forward propagation of uncertainty to reduce the cost of approximating the posterior density. Specifically, we investigate goal-oriented adaptive response surface approximation methods to concentrate high-fidelity model evaluations in regions of the parameters space that both significantly contribute to the uncertainty in specified QoI and are informed by the available data. Numerical results are presented to demonstrate the fact that our approach is consistent with the model and the data, to compare our approach with the statistical Bayesian approach, and to demonstrate the effectiveness of the goal-oriented adaptivity.

BIO: Tim Wildey is a Principle Member of Technical Staff at Sandia National Laboratories in Albuquerque, New Mexico, USA. He currently works on verification, validation, uncertainty quantification and multi-scale modeling in the Optimization and Uncertainty Quantification department within the Center for Computing Research. He earned his PhD in Mathematics from Colorado State University in 2007 working on adjoint-based a posteriori error analysis for coupled multi-physics problems. From 2007 to 2010 he was an ICES postdoctoral fellow at the University of Texas at Austin working in the Center for Subsurface Modeling on multi-scale methods and advanced uncertainty quantification algorithms for flow and mechanics in porous media.