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Numerical differentiation from a viewpoint of regularization theory


Authors: Shuai Lu and Sergei V. Pereverzev
Journal: Math. Comp. 75 (2006), 1853-1870
MSC (2000): Primary 65D25; Secondary 65J20
DOI: https://doi.org/10.1090/S0025-5718-06-01857-6
Published electronically: May 15, 2006
MathSciNet review: 2240638
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Abstract: In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.


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Additional Information

Shuai Lu
Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, A-4040 Linz, Austria
Email: shuai.lu@oeaw.ac.at

Sergei V. Pereverzev
Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, A-4040 Linz, Austria
Email: sergei.pereverzyev@oeaw.ac.at

DOI: https://doi.org/10.1090/S0025-5718-06-01857-6
Keywords: Numerical differentiation, adaptive regularization, unknown smoothness, finite-difference methods, Tikhonov regularization
Received by editor(s): November 3, 2004
Received by editor(s) in revised form: April 19, 2005
Published electronically: May 15, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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