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Approximation of Bayesian Inverse Problems for PDEs

Cotter, S. L. and Dashti, M. and Stuart, A. M. (2010) Approximation of Bayesian Inverse Problems for PDEs. SIAM Journal on Numerical Analysis, 48 (1). pp. 322-345. ISSN 0036-1429.

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Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier–Stokes equation from Lagrangian and Eulerian observations, respectively.

Item Type:Article
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URLURL TypeDescription Paper
Additional Information:© 2010 Society for Industrial and Applied Mathematics. Received by the editors September 10, 2009; accepted for publication (in revised form) February 19, 2010; published electronically April 2, 2010. This research was supported by the EPSRC, ERC, and ONR.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Subject Keywords:inverse problem, Bayesian, Stokes flow, data assimilation, Markov chain–Monte Carlo
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ81
Issue or Number:1
Classification Code:AMS subject classifications. 35K99, 65C50, 65M32
Record Number:CaltechAUTHORS:20160804-170531840
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69458
Deposited By: Linda Taddeo
Deposited On:08 Aug 2016 17:55
Last Modified:03 Oct 2019 10:22

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