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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 5, Pages 767–783 (Mi zvmmf287)

On the total-variation convergence of regularizing algorithms for ill-posed problems

A. S. Leonov

Moscow Engineering Physics Institute, Kashirskoe sh. 31, Moscow, 115409, Russia

Abstract: It is well known that ill-posed problems in the space $V[a,b]$ of functions of bounded variation cannot generally be regularized and the approximate solutions do not converge to the exact one with respect to the variation. However, this convergence can be achieved on separable subspaces of $V[a,b]$. It is shown that the Sobolev spaces $W_1^m[a,b]$, $m\in\mathbb N$ can be used as such subspaces. The classes of regularizing functionals are indicated that guarantee that the approximate solutions produced by the Tikhonov variational scheme for ill-posed problems converge with respect to the norm of $W_1^m[a,b]$. In turn, this ensures the convergence of the approximate solutions with respect to the variation and the higher order total variations.

Key words: ill-posed problems, regularizing algorithms, space of functions of bounded variation, Sobolev space.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:5, 732–747

Bibliographic databases:

UDC: 519.642.8

Citation: A. S. Leonov, “On the total-variation convergence of regularizing algorithms for ill-posed problems”, Zh. Vychisl. Mat. Mat. Fiz., 47:5 (2007), 767–783; Comput. Math. Math. Phys., 47:5 (2007), 732–747

Citation in format AMSBIB
\Bibitem{Leo07} \by A.~S.~Leonov \paper On the total-variation convergence of regularizing algorithms for ill-posed problems \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2007 \vol 47 \issue 5 \pages 767--783 \mathnet{http://mi.mathnet.ru/zvmmf287} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2378657} \elib{http://elibrary.ru/item.asp?id=9535249} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 5 \pages 732--747 \crossref{https://doi.org/10.1134/S0965542507050028} \elib{http://elibrary.ru/item.asp?id=13543322} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34249692374} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Poeschl Ch., Resmerita E., Scherzer O., “Discretization of variational regularization in Banach spaces”, Inverse Problems, 26:10 (2010), 105017
2. Koshev N.A., Orlikovskii N.A., Rau E.I., Yagola A.G., “Reshenie obratnoi zadachi vosstanovleniya signala elektronnogo mikroskopa v rezhime otrazhennykh elektronov na mnozhestve funktsii ogranichennoi variatsii”, Vychislitelnye metody i programmirovanie: novye vychislitelnye tekhnologii, 12:1 (2011), 362–367
3. A. S. Leonov, “Higher-order total variations for functions of several variables and their application in the theory of ill-posed problems”, Proc. Steklov Inst. Math. (Suppl.), 280, suppl. 1 (2013), 119–133
4. Vasin V.V., “Regularization of Ill-Posed Problems By Using Stabilizers in the Form of the Total Variation of a Function and Its Derivatives”, J. Inverse Ill-Posed Probl., 24:2 (2016), 149–158
5. Vasin V.V., Belyaev V.V., “Modification of the Tikhonov Method Under Separate Reconstruction of Components of Solution With Various Properties”, Eurasian J. Math. Comput. Appl., 5:2 (2017), 66–79
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