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### Open Access Journals

IPI

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.

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