An overview on convergence rates for Tikhonov regularization methods for non-linear operators

Abstract

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. The first convergence rates results for non-linear problems have been developed by Engl, Kunisch and Neubauer in 1989 [Inverse Problems 5: 523–540, 1989]. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004 [Inverse Problems 20: 1411–1421, 2004], more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and Scherzer [Inverse Problems 22: 801–814, 2006] presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the 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 on the operator do not necessarily affect the convergence rate negatively. We have taken this observation and weakened the smoothness assumptions on the operator and have proved a novel convergence rate result published in [Hofmann, Kaltenbacher, Pöschl, Scherzer, Inverse Problems 23: 987–1010, 2007]. The most significant difference of this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before, which is necessary due to the lack of smoothness assumptions on F .