Lavrentiev's regularization method in Hilbert spaces revisited

Bernd  Hofmann; Barbara  Kaltenbacher; Elena  Resmerita

doi:10.3934/ipi.2016019

Article Contents

2016, Volume 10, Issue 3: 741-764. Doi: 10.3934/ipi.2016019

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Lavrentiev's regularization method in Hilbert spaces revisited

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

Received: May 31, 2015

Revised: February 29, 2016

Published: July 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 65J22; Secondary: 45Q05, 35R30, 47H05.

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