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Approximation of Bayesian inverse problems for PDEs

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Cotter, Simon L., Dashti, M. and Stuart, A. M.. (2010) Approximation of Bayesian inverse problems for PDEs. SIAM Journal on Numerical Analysis, Vol.48 (No.1). pp. 322-345. ISSN 0036-1429

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Official URL: http://dx.doi.org/10.1137/090770734

Abstract

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:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Differential equations, Partial -- Improperly posed problems, Inverse problems (Differential equations), Bayesian statistical decision theory, Stokes flow
Journal or Publication Title:	SIAM Journal on Numerical Analysis
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1429
Date:	2010
Volume:	Vol.48
Number:	No.1
Number of Pages:	24
Page Range:	pp. 322-345
Identification Number:	10.1137/090770734
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC)
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URI:	http://wrap.warwick.ac.uk/id/eprint/3313