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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. ISSN 02665611

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Official URL: http://dx.doi.org/10.1088/02665611/28/4/045003
Abstract
We present a parametric deterministic formulation of Bayesian inverse problems with an input parameter from infinitedimensional, 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 

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:  02665611 
Date:  April 2012 
Volume:  Vol.28 
Number:  No.4 
Number of Pages:  33 
Page Range:  045003 
Identification Number:  10.1088/02665611/28/4/045003 
Status:  Not Peer Reviewed 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) 
References:  [1] R. Andreev, Ch. Schwab and A.M. Stuart. In preparation. 
URI:  http://wrap.warwick.ac.uk/id/eprint/48173 
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