Abstract

This talk presents the new trends and recent developments of Tikhonov regularizations in Hilbert and Banach settings, which is one of the major tools to obtain stable approximations of ill-posed problems. In particular, we propose and analyze variational source conditions (VSC) for the Tikhonov regularization method with Lp-penalties applied to an ill-posed operator equation in a Banach space. Our analysis is built on the celebrated Littlewood-Paley theory and the concept of (Rademacher) R-boundedness. With these two analytical principles, we validate the proposed VSC under a conditional stability estimate in terms of a dual Triebel-Lizorkin-type norm. Some applications in parameter identification are given.