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Besov priors for Bayesian inverse problems

Dashti, Masoumeh and Harris, Stephen J. and Stuart, Andrew (2012) Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 6 (2). pp. 183-200. ISSN 1930-8345. doi:10.3934/ipi.2012.6.183. https://resolver.caltech.edu/CaltechAUTHORS:20160728-153255881

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Abstract

We consider the inverse problem of estimating a function u from noisy, possibly nonlinear, observations. We adopt a Bayesian approach to the problem. This approach has a long history for inversion, dating back to 1970, and has, over the last decade, gained importance as a practical tool. However most of the existing theory has been developed for Gaussian prior measures. Recently Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct Besov prior measures, based on wavelet expansions with random coefficients, and used these prior measures to study linear inverse problems. In this paper we build on this development of Besov priors to include the case of nonlinear measurements. In doing so a key technical tool, established here, is a Fernique-like theorem for Besov measures. This theorem enables us to identify appropriate conditions on the forward solution operator which, when matched to properties of the prior Besov measure, imply the well-definedness and well-posedness of the posterior measure. We then consider the application of these results to the inverse problem of finding the diffusion coefficient of an elliptic partial differential equation, given noisy measurements of its solution.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.3934/ipi.2012.6.183DOIArticle
http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=7425PublisherArticle
https://arxiv.org/abs/1105.0889arXivDiscussion Paper
Additional Information:© 2012 American Institute of Mathematical Sciences. Communicated by Matti Lassas. Received May 2011; revised March 2012. MD and AMS are grateful to the EPSRC (UK) and ERC for financial support.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Subject Keywords:nverse problems, Bayesian approach, Besov measures, elliptic inverse problems, Fernique-like theorem
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ94
Issue or Number:2
Classification Code:Primary: 60H30, 60G50, 60G15; Secondary: 35J99
DOI:10.3934/ipi.2012.6.183
Record Number:CaltechAUTHORS:20160728-153255881
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160728-153255881
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69293
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:29 Jul 2016 22:34
Last Modified:11 Nov 2021 04:12

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