A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators 论文

摘要

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 first convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While these results apply 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 modification 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 relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence