Past Event: Oden Institute Seminar

A Consistent Bayesian Approach for Stochastic Inverse Problems

Tim M. Wildey, Computer Science Research Institute, Sandia National Laboratories

Abstract

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 measurable 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 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 probability density that is consistent with the model and the data in the sense that the push-forward of the posterior through the model will match the observed density on the data. Our approach requires approximating the push-forward 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. Numerical results are presented to demonstrate the fact that our approach is consistent with the model and the data, and to compare our approach with the statistical Bayesian approach. Bio Tim Wildey is a Senior Member of the Technical Staff at the Computer Science Research Institute at Sandia National Laboratories in Albuquerque, NM. His research interests are finite element and finite volume methods, a posteriori error analysis and estimation, uncertainty quantification, adjoint methods, multiphysics and multiscale problems, operator splitting and decomposition, computational fluid dynamics, geomechanics, flow and transport in porous media, numerical linear algebra, domain decomposition, multilevel and multiscale preconditioners, and parallel computing.