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Inverse problems: A Bayesian perspective

Stuart, A. M. (2010) Inverse problems: A Bayesian perspective. Acta Numerica, 19 . pp. 451-559. ISSN 0962-4929.

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The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems. We demonstrate that, when formulated in a Bayesian fashion, a wide range of inverse problems share a common mathematical framework, and we high- light a theory of well-posedness which stems from this. The well-posedness theory provides the basis for a number of stability and approximation results which we describe. We also review a range of algorithmic approaches which are used when adopting the Bayesian approach to inverse problems. These include MCMC methods, filtering and the variational approach.

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Additional Information:© 2010 Cambridge University Press. The material in this article is developed in greater detail in the lecture notes of Dashti et al.(2010b). These notes are freely available for download from: The author is grateful to his co-authors Masoumeh Dashti and Natesh Pillai for their input into this article. The author also thanks Sergios Agapiou, Andrew Duncan, Stephen Harris, Sebastian Reich and Sebastian Vollmer for numerous comments which improved the presentation, and to Daniella Calvetti and Erkki Somersalo for useful pointers to relevant literature. Finally, the author is grateful to have received financial support from the Engineering and Physical Sciences Research Council (UK), the European Research Council and from the US Office of Naval Research during the writing of this article. This funded research has helped shape much of the material presented.
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Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
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Andrew StuartC15
Record Number:CaltechAUTHORS:20161111-112136150
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Official Citation:Stuart, A.M. (2010) ‘Inverse problems: A Bayesian perspective’, Acta Numerica, 19, pp. 451–559. doi: 10.1017/S0962492910000061.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71949
Deposited By: Linda Taddeo
Deposited On:15 Nov 2016 00:26
Last Modified:03 Oct 2019 16:12

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