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Zh. Vychisl. Mat. Mat. Fiz.:

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

This article is cited in 5 scientific papers (total in 5 papers)

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
Received: 09.02.2006

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
\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
\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 5
\pages 732--747

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    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    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  mathnet  elib
    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  mathnet  crossref  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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