Journal
JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS
Volume 22, Issue 3, Pages 483-517Publisher
ROCKY MT MATH CONSORTIUM
DOI: 10.1216/JIE-2010-22-3-483
Keywords
Ill-posed problems; inverse problems; noisy right hand side; noisy operator; Tikhonov regularization; Hilbert scales; generalized discrepancy principle; order optimal error bounds; Newton's method; global convergence; monotone convergence
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Funding
- Austrian Fonds Zur Forderung der Wissenchaftlichen Forschung (FWF) [P20235-N18]
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For solving linear ill-posed problems regularization methods are required when the right hand side and the operator are with some noise. In the present paper regularized solutions are obtained by Tikhonov regularization in Hilbert scales and the regularization parameter is chosen by the generalized discrepancy principle. under certain smoothness assumptions we provide order optimal error bounds that characterize the accuracy of the regularized solution. It appears that for getting small error bounds a proper scaling of the penalizing operator B is required. For the computation of the regularization parameter fast algorithms of Newton type are constructed which are based on special transformations. These results extend earlier results where the problem operator is exactly given. Some of our theoretical results are illustrated by numerical experiments.
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