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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 non-destructive testing to satellite geodetic
exploration, operator equations of the form
The problems (1) are ill-posed, in most cases even severely ill-posed, and have to be regularized for solution. Since the problems are usually given in infinite-dimensional spaces they must be additionally discretized. Therefore the well-known 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 ill-posed 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 convolution-type 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 real-world problems.
References:
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