Past Event: Oden Institute Seminar

Computationally efficient Bayesian inference using polynomial chaos expansions

Youssef Marzouk, Professor, MIT, Department of Aeronautics & Astronautics

Abstract

Predictive simulation of complex engineering systems increasingly rests on the interplay of experimental observations with computational models. Key inputs, parameters, or structural aspects of models may be incomplete or unknown, and must be developed from indirect and limited observations. At the same time, quantified uncertainties are needed to qualify computational predictions in the support of design and decision-making. In this context, Bayesian statistics provides a complete foundation for inference from noisy and limited data. Computationally intensive forward models, however, can render a Bayesian approach prohibitive. Polynomial chaos expansions, typically used in the forward propagation of uncertainty, are an extremely useful tool in the inverse context as well. We introduce a stochastic spectral formulation that accelerates the Bayesian solution of inverse problems via rapid evaluation of a surrogate posterior distribution. The posterior is constructed by either stochastic collocation or stochastic Galerkin methods. Theoretical convergence results are verified with several numerical examples---in particular, parameter estimation in transport equations and in chemical kinetic systems. We also extend this approach to the inference of spatially distributed quantities in a hierarchical Bayesian setting, achieving dimensionality reduction via Karhunen-Loeve representations of Gaussian process priors. Finally, we discuss the utility of polynomial chaos expansions in density estimation, formulating a hierarchical Bayesian method for estimating polynomial chaos representations from sparse data. Here, we introduce a reversible-jump Markov chain Monte Carlo scheme that simultaneously traverses polynomial degree and the corresponding spaces of coefficients, thus extending the parameter estimation problem to one of model averaging and model selection. Host: Omar Ghattas Note: Dr. Marzouk will be visiting ICES from July 20 - August 4. Anyone wishing to meet with him should contact Youssef at ymarz@mit.edu. His office will be POB 4.234.