Convergence rates for variational regularization of statistical inverse problems

by Benjamin Sprung
Doctoral thesis
Date of Examination:2019-10-04
Date of issue:2020-05-13
Advisor:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Andreas Neubauer
Persistent Address: http://dx.doi.org/10.53846/goediss-7965

 

 

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Abstract

English

We consider inverse problems with statistical (noisy) data. By applying regularization methods one can approximate the true solution of the inverse problem by a regularized solution. In this thesis we show convergence rates of the regularized solution to the true solution as the noise tends to zero under so called source conditions on the true solution. Recently variational source conditions (VSCs) have become increasingly popular, due to their generality. However, they have the disadvantage that they only give optimal rates for low smoothness of the true solution. For this reason a second order VSC has been introduced in the literature, but it still has the problem that it can only make use of limited smoothness. A major contribution of this work is that we introduce VSCs of arbitrary order on Hilbert spaces which cover convergence rates of arbitrary order. For more general Banach spaces we introduce a third order VSC with which we show the new result that Bregman iterated Tikhonov regularization can have improved convergence rates compared to noniterated Tikonov regularization. A special focus of this work lies on statistical noise models such as Gaussian white noise and Poisson data. We show that also in these settings VSCs can yield order optimal convergence rates in expectation. Further we introduce variants of the second order VSC that yield new higher order convergence rates for Poisson data. In addition to these major contributions this work also contains a new proof of upper and lower bounds for Bregman divergences, which are a crucial tool in our convergence analysis.
Keywords: regularization theory; convergence rates; Gaussian white noise; Poisson data; variational source conditions; Bregman divergence; Bregman iterated Tikhonov regularization
 
Statistik