Analysis and computation for Bayesian inverse problems

Abstract

Many inverse problems involve the estimation of a high dimensional quantity, such as a function, from noisy indirect measurements. These problems have received much study from both classical and statistical directions, with each approach having its own advantages and disadvantages. In this thesis we focus on the Bayesian approach, in which all uncertainty is modelled probabilistically. Recently the Bayesian approach to inversion has been developed in function space. Much of the existing work in the area has been focused on the case when the prior distribution produces samples which are continuous functions, however it is of interest, both in terms of applications and mathematically, to consider cases when these samples are discontinuous. Natural applications are those in which we wish to infer the shape and locations of interfaces between different materials, such as in tomography. In this thesis we consider Bayesian inverse problems in which the unknown function is piecewise continuous or piecewise constant. Based on prior information, the problem is then to infer the discontinuity set, the values the function takes away from the discontinuities, or both simultaneously. These problems are considered both from analytic and computational points of view. In order to ensure numerical robustness, we formulate any algorithms directly on function space before discretizing. This requires a number of technical issues to be considered, such as the equivalence and singularity of measures on such spaces.