Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems

Abstract

In the convergence theory of regularization methods for ill-posed problems, so far deterministic error concepts have dominated, which leads to worst-case error estimates. Since this is sometimes not desirable, we aim at providing a framework for proving convergence rates in the Prokhorov metric for the regularization of ill-posed problems with stochastic noise. This allows us to assess uncertainty in the sense of a confidence region for the probability that the deviation between the exact and regularized solutions stays below a given bound with given probability. We exemplify this method for the special case of Tikhonov regularization for linear ill-posed problems and apply the result to the problem of deblurring an image contaminated by random blurring.