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MAP estimators for piecewise continuous inversion
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Dunlop, Matthew M. and Stuart, A. M. (2016) MAP estimators for piecewise continuous inversion. Inverse Problems, 32 (10). 105003. doi:10.1088/0266-5611/32/10/105003 ISSN 0266-5611.
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Official URL: http://dx.doi.org/10.1088/0266-5611/32/10/105003
Abstract
We study the inverse problem of estimating a field u a from data
comprising a finite set of nonlinear functionals of u a , subject to additive noise; we denote this observed data by y. Our interest is in the reconstruction of piecewise continuous fields u a in which the discontinuity set is described by a finite number of geometric parameters a. Natural applications include groundwater flow and electrical impedance tomography. We take a Bayesian approach, placing a prior distribution on u a and determining the conditional distribution on u a given the data y. It is then natural to study maximum a posterior (MAP) estimators. Recently (Dashti et al 2013 Inverse roblems 29 095017) it has been shown that MAP estimators can
be characterised as minimisers of a generalised Onsager-Machlup functional, in the case where the prior measure is a Gaussian random field. We extend this theory to a more general class of prior distributions which allows for piecewise continuous fields. Specifically, the prior field is assumed to be piecewise Gaussian with random
interfaces between the different Gaussians defined by a finite number of parameters.
We also make connections with recent work on MAP estimators for linear problems and possibly non-Gaussian priors (Helin, Burger 2015 Inverse Problems 31 085009) which employs the notion of Fomin derivative. In showing applicability of our theory we focus on the groundwater flow and EIT models, though the theory holds more generally. Numerical experiments are implemented for the groundwater flow model, demonstrating the feasibility of
determining MAP estimators for these piecewise continuous models, but also that the geometric formulation can lead to multiple nearby (local) MAP estimators. We relate these MAP estimators to the behaviour of output from MCMC samples of the posterior, obtained using a state-of-the-art function space Metropolis-Hastings method.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Inverse problems (Differential equations), Groundwater flow -- Mathematical models, Bayesian statistical decision theory, Geometric probabilities | ||||||||
Journal or Publication Title: | Inverse Problems | ||||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||||
ISSN: | 0266-5611 | ||||||||
Official Date: | 8 August 2016 | ||||||||
Dates: |
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Volume: | 32 | ||||||||
Number: | 10 | ||||||||
Article Number: | 105003 | ||||||||
DOI: | 10.1088/0266-5611/32/10/105003 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 19 July 2016 | ||||||||
Date of first compliant Open Access: | 16 December 2016 | ||||||||
Funder: | University of Warwick. Mathematics and Statistics Doctoral Training Centre, Engineering and Physical Sciences Research Council (EPSRC), Great Britain. Office for Nuclear Regulation (ONR) | ||||||||
Grant number: | H023364/1, EP/K000128/1 (EPSRC) |
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