Bayesian analysis in inverse problems

In this paper, we consider some statistical aspects of inverse problems. using Bayesian analysis, particularly estimation and hypothesis-testing questions for parameterdependent differential equations. We relate Bayesian maximum likelihood Io Tikhonw regularization. and we apply the expectatian-minimization (F-M) algorithm to the problem of setting regularization levels. Further, we compare Bayesian resulls with those of a classical statistical approach, through consistency and asymptotic normality. A numerical example illustrates the application of Bayesian techniques. In many cases one i s interested in parameters which are infinite dimensional (e.g. functions). Bayesian techniques offer a sound theoretical and computational paradigm, through probabilily measures on Banach space. We develop a framework for infinite dimensional Bayesian analysis, including convergence of approximalions required to perform inference tasks computationally.