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

Author(s):

Robert Plato
Peter Mathé
Bernd Hofmann

Title:

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

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 03, 2016

Mathematics Subject Classification:

    47A52 []
    65J20 []
    47H06 []

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, {\it Fractional powers of dissipative operators}. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on 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 nonnegative selfadjoint operators.

Keywords:

Linear ill-posed problems, regularization, accretive operators, source conditions, convergence rates

Language:

English

Publication time:

3/2016