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Uncertainty quantification and weak approximation of an elliptic inverse problem
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Dashti, M. and Stuart, A. M.. (2011) Uncertainty quantification and weak approximation of an elliptic inverse problem. SIAM Journal on Numerical Analysis, Vol.49 (No.6). pp. 25242542. ISSN 00361429

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Official URL: http://dx.doi.org/10.1137/100814664
Abstract
We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimension $d \le 3$, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen–Loève expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a form of well posedness of the inverse problem. Determining the posterior measure, given the data, solves the problem of uncertainty quantification for this inverse problem. In practice the posterior measure must be approximated in a finite dimensional space. We quantify the errors incurred by employing a truncated Karhunen–Loève expansion to represent this meausure. In particular we study weak convergence of a general class of locally Lipschitz functions of the log permeability, and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance, under refinement of the finitedimensional Karhunen–Loève truncation.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Darcy's law, Inverse problems (Differential equations), Differential equations, Elliptic 
Journal or Publication Title:  SIAM Journal on Numerical Analysis 
Publisher:  Society for Industrial and Applied Mathematics 
ISSN:  00361429 
Official Date:  2011 
Volume:  Vol.49 
Number:  No.6 
Page Range:  pp. 25242542 
Identification Number:  10.1137/100814664 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC) 
References:  [1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, London, 1975. 
URI:  http://wrap.warwick.ac.uk/id/eprint/44116 
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