Geodesic
The finite difference approximation for the Tikhonov regularization method of the n-th order
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 4 (2015) no. 1, pp. 86-98

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This article is a natural extension of the work by A.N. Tikhonov, where the idea of a finitedimensional approximation of the regularization problem was first formulated. However, the conditions, offered for operators, are difficult to verify. In the present work we offer other conditions, which are easier to use in practice, and use it to prove the theorem of convergence of the finitedimensional approximation for the Tikhonov regularization method. Application of the described method is demonstrated by the example with the Fredholm equation of the first kind.
Keywords: inverse problem, regularization, finite difference approximation, ill-posed problem, integral equation. 
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     title = {The finite difference approximation for the {Tikhonov} regularization method of the n-th order},
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V. P. Tanana; S. I. Belkov. The finite difference approximation for the Tikhonov regularization method of the n-th order. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 4 (2015) no. 1, pp. 86-98. http://geodesic.mathdoc.fr/item/VYURV_2015_4_1_a7/