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Sparse deterministic approximation of Bayesian inverse problems
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Schwab, Ch. (Christoph) and Stuart, A. M. (2012) Sparse deterministic approximation of Bayesian inverse problems. Inverse Problems, Vol.28 (No.4). 045003. doi:10.1088/0266-5611/28/4/045003
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Official URL: http://dx.doi.org/10.1088/0266-5611/28/4/045003
Abstract
We present a parametric deterministic formulation of Bayesian inverse problems with an input parameter from infinite-dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. The first step in this process is to estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number N of unknowns appearing in the parametric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise.
| Item Type: | Journal Article | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science > Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Bayesian statistical decision theory, Banach spaces, Differential equations, Partial | ||||
| Journal or Publication Title: | Inverse Problems | ||||
| Publisher: | Institute of Physics Publishing Ltd. | ||||
| ISSN: | 0266-5611 | ||||
| Official Date: | April 2012 | ||||
| Dates: |
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| Volume: | Vol.28 | ||||
| Number: | No.4 | ||||
| Number of Pages: | 33 | ||||
| Page Range: | 045003 | ||||
| DOI: | 10.1088/0266-5611/28/4/045003 | ||||
| Status: | Not Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Funder: | Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [Swiss National Science Foundation] (SNSF), Seventh Framework Programme (European Commission) (FP7), Engineering and Physical Sciences Research Council (EPSRC) | ||||
| Grant number: | AdG247277 (FP7) |
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