A Convergence Rates Result in Banach Spaces with Non-Smooth Operators
There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The rst convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While this result applies for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita et al. presented a modi cation of the convergence rates result of Burger and Osher which turns out a complete generalization of the rates result of Engl et. al. In all these papers relative strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions of the operator do not necessarily a ect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most signi cant di erence in this result to the previous ones is that the source condition is formulated as variational inequality and not as an equation as before. We reconsider an example from nance and a phase retrieval problem, both studied in the literature before.