# American Institute of Mathematical Sciences

August  2016, 10(3): 741-764. doi: 10.3934/ipi.2016019

## Lavrentiev's regularization method in Hilbert spaces revisited

 1 Technische Universität Chemnitz, Reichenhainer Str. 41, 09111 Chemnitz, Germany 2 Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  June 2015 Revised  March 2016 Published  August 2016

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.
Citation: Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems & Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019
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##### References:
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