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Optimal rates for Lavrentiev regularization with adjoint source conditions

Authors: Robert Plato, Peter Mathé and Bernd Hofmann
Journal: Math. Comp. 87 (2018), 785-801
MSC (2010): Primary 65N20; Secondary 45Q05
Published electronically: June 13, 2017
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Abstract: There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive, then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato [J. Math. Soc. Japan 13(1961), no. 3, 247-274], we establish power type convergence rates for this case. By measuring the optimality of such rates in terms of limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of positive semidefinite selfadjoint operators.

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Additional Information

Robert Plato
Affiliation: Department of Mathematics, University of Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany

Peter Mathé
Affiliation: Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany

Bernd Hofmann
Affiliation: Faculty of Mathematics, TU Chemnitz, 09107 Chemnitz, Germany

Received by editor(s): March 14, 2016
Received by editor(s) in revised form: August 4, 2016, and October 23, 2016
Published electronically: June 13, 2017
Additional Notes: The research of the third author was supported by the German Research Foundation (DFG) under grant HO 1454/8-2.
Article copyright: © Copyright 2017 American Mathematical Society

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