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Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors
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Sullivan, T. J. (2017) Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors. Proceedings in Applied Mathematics and Mechanics, 17 (1). pp. 871-874. doi:10.1002/pamm.201710402 ISSN 1617-7061.
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Official URL: http://dx.doi.org/10.1002/pamm.201710402
Abstract
The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite‐dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well‐posed BIPs in infinite‐dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others. This framework has the advantage of ensuring uniformly well‐posed inference problems independently of the finite‐dimensional discretisation used for numerical solution. Recently, this framework has been extended to the case of a heavy‐tailed prior measure in the family of stable distributions, such as an infinite‐dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square‐integrable random variables can be used to sample such measures on quasi‐Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering Faculty of Science, Engineering and Medicine > Science > Mathematics |
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Journal or Publication Title: | Proceedings in Applied Mathematics and Mechanics | ||||||
Publisher: | Wiley - V C H Verlag GmbH & Co. KGaA | ||||||
ISSN: | 1617-7061 | ||||||
Official Date: | December 2017 | ||||||
Dates: |
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Volume: | 17 | ||||||
Number: | 1 | ||||||
Page Range: | pp. 871-874 | ||||||
DOI: | 10.1002/pamm.201710402 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Reuse Statement (publisher, data, author rights): | Free access from Publisher's site | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Description: | Special Issue: 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017; Editors: C. Könke, Weimar, and C. Trunk, Ilmenau |
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