Düvelmeyer, Dana ; Hofmann, Bernd ; Yamamoto, Masahiro : Range inclusions and approximate source conditions with general benchmark functions

Author(s):

Düvelmeyer, Dana
Hofmann, Bernd
Yamamoto, Masahiro

Title:

Range inclusions and approximate source conditions with general benchmark functions

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 2, 2007

Mathematics Subject Classification:

47A52			[ Ill-posed problems, regularization ]
65J20			[ Improperly posed problems; regularization ]

Abstract:

The paper is devoted to the analysis of ill-posed operator equations $Ax=y$ with injective linear operator $A$ and solution $x_0$ in a Hilbert space setting. We present some new ideas and results for finding convergence rates in Tikhonov regularization based on the concept of approximate source conditions by means of using distance functions with a general benchmark. For the case of compact operator $A$ and benchmark functions of power-type we can show that there is a one-to-one correspondence between the maximal power-type decay rate of the distance function and the best possible H\"older exponent for the noise-free convergence rate in Tikhonov regularization. As is well-known this exponent coincides with the supremum of exponents in power-type source conditions. The main theorem of this paper is devoted to the impact of range inclusions under the smoothness assumption that $x_0$ is in the range of some positive self-adjoint operator $G$. It generalizes a convergence rate result proven for compact $G$ in [12] to the case of general operators $G$ with non-closed range.

Keywords:

linear ill-posed problems, Tikhonov regularization, distance functions, convergence rates, Hölder exponent, index function, qualification, source condition, range inclusion

Language:

English

Publication time:

1 / 2007