Regularization in Banach spaces - convergence rates theory

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Motivated by their successful application in image restoring and sparsity reconstruction this manuscript deals with regularization theory of linear and nonlinear inverse and ill-posed problems in Banach space settings. Whereas regularization in Hilbert spaces has been widely studied in literature for a long period the developement and investigation of regularization methods in Banach spaces have become a field of modern research. The manuscript is twofolded. The first part deals with convergence rates theory for Tikhonov regularization as classical regularization method. In particular, generalizations of well-established results in Hilbert spaces are presented in the Banach space situation. Since the numerical effort of Tikhonov regularization in applications is rather high iterative approaches were considered as alternative regularization variants in the second part. In particular, two Gradient-type methods were presented and their behaviour concerning convergence and stability is investigated. For one of the methods, additionally, a convergence rates result is formulated. All the theoretical results are illustrated by some numerical examples.