Besov priors for Bayesian inverse problems

We consider the inverse problem of estimating a function ufrom noisy, possibly nonlinear, observations.We adopt a Bayesian approach to the problem. Thisapproach 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 beendeveloped 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 theseprior measures to study linear inverse problems. In thispaper we build on this development of Besov priors to includethe case of nonlinear measurements. In doing so a key technical tool,established here, is a Fernique-like theorem forBesov measures. This theorem enables us toidentify appropriate conditions on the forward solutionoperator which, when matched to properties of the priorBesov measure, imply the well-definedness and well-posednessof the posterior measure. We then consider the application of theseresults to the inverse problem of finding the diffusion coefficientof an elliptic partial differential equation, given noisymeasurements of its solution.