Numerical differentiation from a viewpoint of regularization theory |
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| Authors: | Shuai Lu Sergei V. Pereverzev. |
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| Affiliation: | Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, A-4040 Linz, Austria ; Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, A-4040 Linz, Austria |
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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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| Keywords: | Numerical differentiation adaptive regularization unknown smoothness finite-difference methods Tikhonov regularization |
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