Besov priors for Bayesian inverse problems

By Masoumeh Dashti, Stephen Harris, Andrew M. Stuart. 2011

130 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39d7c7201d497be2d95bb5aa9c45369017363122

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Citations in the corpus

Bayesian Probabilistic Numerical Methods (2017, methodology): “More general priors, such as Besov priors (Dashti et al., 2012) and Cauchy priors (Sullivan, 2016) are less easily accessed.”
Data Assimilation: A Mathematical Introduction (2015, methodology): “This idea is used for data assimilation problems arising in fluid mechanics in [28], for inverse problems arising in subsurface geophysics in [36, 34] and described more generally in [106].”
MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster (2012, background): “3 is shown to satisfy these assumptions in [13], again for approximate choice of X.”
Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions (2011, methodology): “Particularly, the Besov priors of [18] provide an interesting class of such reference measures, and the paper [13] provides a machinery for analyzing change of measure from the Besov prior, analogous to that used here in the Gaussian case.”
Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions (2011, background): “Other classes of target measure, such as those arising from Besov prior measures [28, 14], or the uniform measures in [47], would also provide interesting applications.”
The Bayesian Approach to Inverse Problems (2013, methodology): “5, Bog98] and are also discussed in [Stu10]. • Subsection 3.4. The elliptic inverse problem with the uniform prior is studied in [SS12]. A Gaussian 33 prior is adopted in [DS11], and a Besov prior in [DHS12]. • Subsection 4.1. Relationships between the Hellinger distance on probability measures, and”