Hofmann, Bernd : The potential for ill-posedness of multiplication operators occurring in inverse problems

Author(s):

Hofmann, Bernd

Title:

The potential for ill-posedness of multiplication operators occurring in inverse problems

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 17, 2004

Mathematics Subject Classification:

65J20			[ Improperly posed problems; regularization ]
47H30			[ Particular nonlinear operators ]
47B33			[ Composition operators ]
65R30			[ Improperly posed problems ]
47B06			[ Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators ]
45C05			[ Eigenvalue problems ]
45P05			[ Integral operators ]

Abstract:

In this paper, we show the restricted influence of non-compact multiplication operators mapping in $L^2(0,1)$ occurring in linear ill-posed operator equations and in the linearization of nonlinear ill-posed operator equations with compact forward operators. We give examples of nonlinear inverse problems in natural science and stochastic finance that can be written as nonlinear operator equations for which the forward operator is a composition of a simple linear integration operator and a nonlinear Nemytskii operator. Hence, the Frechet derivative of such a forward operator is a composition of integration and multiplication operators. It is shown for power type functions and conjectured for a wider class of weight functions with essential zeros that the unbounded inverse of the injective multiplication operator does not influence the (local) degree of ill-posedness of inverse problems under consideration. In a more general Hilbert space setting, we investigate the role of approximate source conditions in the method of Tikhonov regularization. We introduce adistance function measuring the violation of canonical source conditions and derive convergence rates based on that functions.

Keywords:

inverse problems, linear and nonlinear ill-posed problems, singular values, degree of ill-posedness, multiplication operator, Nemytskii operator, Tikhonov regularization, Frechet derviative, convergence rates, source conditions

Language:

English

Publication time:

11 / 2004