Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 228, Issue 6, Pages 1862-1902Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2008.11.024
Keywords
Inverse problems; Bayesian inference; Dimensionality reduction; Polynomial chaos; Markov chain Monte Carlo; Galerkin projection; Gaussian processes; Karhunen-Loeve expansion; RKHS
Funding
- Sandia National Laboratories Truman Fellowship in National Security Science and Engineering
- Sandia Corporation
- US Department of Energy [DE-AC04-94AL85000]
- Office of Advanced Scientific Computing Research (ASCR)
- Office of Basic Energy Sciences (BES)
- Division of Chemical Sciences, Geosciences, and Biosciences
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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-Loeve 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. (C) 2008 Elsevier Inc. All rights reserved.
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