


[Contents]  [Index] 
Collaborator: G. Bruckner
Cooperation with: J. Cheng (Fudan University, Shanghai, China), S.V. Pereverzev (National Academy of Sciences of Ukraine, Kiev)
Description:
In the mathematical treatment of inverse problems ranging from
tomography over nondestructive testing to satellite geodetic
exploration, operator equations of the form
The problems (1) are illposed, in most cases even severely illposed, and have to be regularized for solution. Since the problems are usually given in infinitedimensional spaces they must be additionally discretized. Therefore the wellknown method of regularization by discretization seems to be natural and is frequently used. Moreover, because of the lack of information about the singular values of A and the smoothness of the solution x, for constructing regularization procedures, the a posteriori parameter choice by discrepancy principles has been proposed. Recently, T. Hohage [3] considered discrepancy principles with respect to Tikhonov's regularization, while the investigations of B. Kaltenbacher [4] referred to the moderately illposed case.
In the present project, the idea of regularization by discretization is continued with a new strategy of a posteriori parameter choice that assumes only estimates for the singular values and the smoothness of the solution (cf. [2]). The algorithm is not based on a discrepancy principle and selects a relevant discretized solution from a number of discretized solutions for subsequent discretization levels. The selected approximate solution has an optimal order of accuracy. Moreover, the algorithm is applied to a problem with a logarithmic convolutiontype operator (cf. [1]), for which the assumptions can be verified. Further work is needed to apply this method to nondestructive testing, quality control of grating devices or other realworld problems.
References:



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