Variational regularization theory for sparsity promoting wavelet regularization

Miller, Philip

dc.contributor.advisor	Hohage, Thorsten Prof. Dr.
dc.contributor.author	Miller, Philip
dc.date.accessioned	2022-01-31T14:46:31Z
dc.date.available	2022-02-07T00:50:04Z
dc.date.issued	2022-01-31
dc.identifier.uri	http://dx.doi.org/10.53846/goediss-5
dc.language.iso	eng	de
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subject.ddc	510	de
dc.title	Variational regularization theory for sparsity promoting wavelet regularization	de
dc.type	doctoralThesis	de
dc.contributor.referee	Hohage, Thorsten Prof. Dr.
dc.date.examination	2021-08-05	de
dc.description.abstracteng	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.	de
dc.contributor.coReferee	Werner, Frank Prof. Dr.
dc.subject.eng	regularization theory	de
dc.subject.eng	wavelet regularization	de
dc.subject.eng	l1 regularization	de
dc.subject.eng	variational source condition	de
dc.subject.eng	inverse problems	de
dc.subject.eng	convex regularization	de
dc.subject.eng	converse result	de
dc.subject.eng	oversmoothing	de
dc.subject.eng	optimal convergence rates	de
dc.identifier.urn	urn:nbn:de:gbv:7-ediss-13852-3
dc.affiliation.institute	Fakultät für Mathematik und Informatik	de
dc.subject.gokfull	Mathematics (PPN61756535X)	de
dc.description.embargoed	2022-02-07	de
dc.identifier.ppn	1795491493

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Fakultät für Mathematik und Informatik (inkl. GAUSS) [520]

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