Abstract

In the talk we give an overview on variational regularization methods in Banach spaces. The theory is developed based on the classical theory of regularization methods in Hilbert spaces. An essential ingredient of such are source conditions, which will be generalized to the Banach spaces setting. After that the example of sparsity regularization, which has proven to be a powerful tool in imaging, will be analyzed in this framework. One of the powerful results proven by Candes et al is a linear convergence rate result. Such a result can be obtained (even in an infinite dimensional setting) from variational regularization theory in Banach spaces as well. Finally we present some applications to Radar imaging, and, if time allows, to Photoacoustic Imaging. This is joint work with M. Grasmair, M. Haltmeier, C. Poschl and E. Resmerita.