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Quasi-solution of linear inverse problems in non-reflexive Banach spaces

  • Christian Clason ORCID logo EMAIL logo and Andrej Klassen
Published/Copyright: August 7, 2018

Abstract

We consider the method of quasi-solutions (also referred to as Ivanov regularization) for the regularization of linear ill-posed problems in non-reflexive Banach spaces. Using the equivalence to a metric projection onto the image of the forward operator, it is possible to show regularization properties and to characterize parameter choice rules that lead to a convergent regularization method, which includes the Morozov discrepancy principle. Convergence rates in a suitably chosen Bregman distance can be obtained as well. We also address the numerical computation of quasi-solutions to inverse source problems for partial differential equations in L(Ω) using a semi-smooth Newton method and a backtracking line search for the parameter choice according to the discrepancy principle. Numerical examples illustrate the behavior of quasi-solutions in this setting.

MSC 2010: 47A52; 65J20; 49M15

Award Identifier / Grant number: CL 487/1-1

Funding statement: This work was supported by the German Science Foundation (DFG) under grant CL 487/1-1.

Acknowledgements

The authors wish to thank Barbara Kaltenbacher for helpful remarks and discussions.

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Received: 2018-03-28
Revised: 2018-06-27
Accepted: 2018-07-05
Published Online: 2018-08-07
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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