Variational regularization theory for sparsity promoting wavelet regularization

by Philip Miller
Doctoral thesis
Date of Examination:2021-08-05
Date of issue:2022-01-31
Advisor:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Frank Werner
Persistent Address: http://dx.doi.org/10.53846/goediss-5

 

 

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Abstract

English

In many scientific and industrial applications, the quantity of interest is not what is directly observed, but is instead a parameter which has a causal effect on experimental measurements. To obtain the desired unknown quantity, one must use an inverse transform on the data. The main challenge in such an inverse problem is that these unknowns may not continuously depend on the observations, and as a result, the effects of noise in data are magnified in the inverted results. To obtain stable approximations of the desired parameters from noisy observations, regularization methods are used. This thesis contributes to the mathematical analysis of generalized Tikhonov regularization, and in particular sparsity promoting Tikhonov regularization, which are popular examples of regularization methods. Using variational source conditions as an intermediate step, order optimal upper bounds on the reconstruction error are shown for sparsity promoting wavelet regularization under smoothness assumptions given by Besov spaces. The framework includes practically relevant forward operators, such as the Radon transform, and some nonlinear inverse problems in differential equations with distributed measurements. In numerical simulations for a parameter identification problem in a differential equation it is demonstrated that these theoretical results correctly predict convergence rates for piecewise smooth unknown coefficients.
Keywords: regularization theory; wavelet regularization; l1 regularization; variational source condition; inverse problems; convex regularization; converse result; oversmoothing; optimal convergence rates
 
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