Ill-Posed Problems: Extendíng the Deterministic Theory to a Stochastic Setup

Ill-posed problems are sensitive to noise in the measurements and must be regularized to achieve meaningful solutions. An important tool for assessing the quality of available regularization methods is a convergence rate analysis. The aim of this work is to develop a framework that allows incorporating stochastic effects into the analysis, including randomness of underlying operator equations and stochasticity of exact solutions. The main instrument in the new approach are the metrics of Ky Fan and Prokhorov.
The central part of this work is concerned with linear and nonlinear inverse problems and their extension to a genuine stochastic setup. For obtaining the new results, stochastic source conditions are required. These are common smoothness conditions on the desired solution, combined with corresponding probabilities. Numerous discussions of new theorems and conditions are included to elucidate the appearing decay conditions. An extensive numerical study on the example of a nonlinear Hammerstein equation completes this discourse.
In a concluding chapter, we point out links to two other concepts. We show that the new framework also allows the comparison of existing probability estimates with deterministic results. Furthermore, we demonstrate that, using the metrics of Ky Fan and Prokhorov, also for the Bayesian approach quantitative convergence results can be derived. Thus, using these metrics, also in this important field it is possible to answer questions concerning convergence and convergence speeds.