Böttcher, Albrecht ; Hofmann, Bernd ; Tautenhahn, Ulrich ; Yamamoto, Masahiro : Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

Author(s):

Böttcher, Albrecht
Hofmann, Bernd
Tautenhahn, Ulrich
Yamamoto, Masahiro

Title:

Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

Electronic source:

application/pdf
application/postscript

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 9, 2005

Mathematics Subject Classification:

47A52			[ Ill-posed problems, regularization ]
65J20			[ Improperly posed problems; regularization ]
65R30			[ Improperly posed problems ]

Abstract:

The paper is concerned with ill-posed operator equations $Ax=y$ where $A:X \to Y$ is an injective bounded linear operator with non-closed range $R(A)$ and $X$ and $Y$ are Hilbert spaces. The solution $x=x^\dagger$ is assumed to be in the range $R(G)$ of some selfadjoint strictly positive bounded linear operator $G:X \to X$. Under several assumptions on $G$, such as $G=\varphi(A^*A)$ or more generally $R(G) \subset R(\varphi(A^*A))$, inequalities of the form $\rho^2(G) \le A^*A$, or range inclusions $R(\rho(G)) \subset R(|A|)$, convergence rates for the regularization error $\|x_{\alpha} - x^\dagger\|$ of Tikhonov regularization are established. We also show that part of our assumptions automatically imply so-called source conditions. The paper contains a series of new results but also intends to uncover cross-connections between the different kinds of smoothness conditions that have been discussed in the literature on convergence rates for Tikhonov regularization.

Keywords:

linear ill-posed problems, Tikhonov regularization, convergence rates, smoothness conditions, index functions, operator monotone functions, range inclusions

Language:

English

Publication time:

6 / 2005