Optimal rates for Lavrentiev regularization with adjoint source conditions

Plato, Robert; Mathé, Peter; Hofmann, Bernd

doi:10.1090/mcom/3237

Optimal rates for Lavrentiev regularization with adjoint source conditions
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by Robert Plato, Peter Mathé and Bernd Hofmann;

Math. Comp. 87 (2018), 785-801

DOI: https://doi.org/10.1090/mcom/3237

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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