Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems
Author(s)Marzouk, Youssef M.; Najm, Habib N.
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We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen–Loève expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data.
DepartmentMassachusetts Institute of Technology. Department of Aeronautics and Astronautics
Journal of Computational Physics
Marzouk, Youssef M., and Habib N. Najm. “Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems.” Journal of Computational Physics 228.6 (2009): 1862-1902.
Author's final manuscript