Geodesic
On the total-variation convergence of regularizing algorithms for ill-posed problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 5, pp. 767-783 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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A. S. Leonov. On the total-variation convergence of regularizing algorithms for ill-posed problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 5, pp. 767-783. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_5_a1/

[1] Tikhonov A. N., Arsenii V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979 | MR

[2] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR

[3] Morozov V. A., Regulyarnye metody resheniya nekorrektno postavlennykh zadach, Nauka, M., 1987 | MR

[4] Tikhonov A. N., Leonov A. C., Yagola A. G., Nelineinye nekorrektnye zadachi, Nauka, M., 1995 | MR

[5] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977 | MR | Zbl

[6] Leonov A. C., “Obobschenie metoda maksimalnoi entropii dlya resheniya nekorrektnykh zadach”, Sibirskii matem. zhurnal, 41:4 (2000), 863–872 | MR | Zbl

[7] Leonov A. C., “Funktsionaly s $H$-svoistvom v prostranstve Soboleva $W_1^1$”, Matem. sb., 195:6 (2004), 121–136 | MR | Zbl

[8] Leonov A. C., “Ob $H$-svoistve funktsionalov v prostranstvakh Soboleva”, Matem. zametki, 77:3 (2005), 378–394 | MR | Zbl

[9] Leonov A. S., “Regularization of ill-posed problems in Sobolev space $W_1^1$”, J. Inv. Ill-Posed Problems, 13:5 (2005), 467–489 | MR

[10] Vinokurov V. A., “O ponyatii regulyarizuemosti razryvnykh otobrazhenii”, Zh. vychisl. matem. i matem. fiz., 11:5 (1971), 1097–1112 | MR | Zbl

[11] Bakushinskii A. B., Goncharskii A. B., Iterativnye metody resheniya nekorrektnykh zadach, Nauka, M., 1989 | MR

[12] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[13] Krasnoselskii M. A., Zabreiko P. P., Pustylnik E. I., Sobolevskii P. E., Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR

[14] Amato U., Hughes W., “Maximum entropy regularization of Fredholm integral equations of the first kind”, Inverse Problems, 7 (1991), 793–808 | DOI | MR | Zbl

[15] Tikhonov A. H., Goncharskii A. B., Stepanov V. V., Yagola A. G., Regulyarizuyuschie algoritmy i apriornaya informatsiya, Nauka, M., 1983 | MR

[16] Nigmatulin B. I., Leonov A. C., Trifonenkov V. P., Khasanov R. Kh., Obratnye zadachi v probleme diagnostiki i upravleniya teplovym rezhimom v korpuse yadernogo reaktora pri tyazhelykh avariyakh, Preprint, NIItsentr po bezopasnosti AES, Elektrogorsk, 1994

[17] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981 | MR

[18] Leonov A. S., “Numerical piecewise-uniform regularization for two-dimensional ill-posed problems”, Inverse Problems, 15 (1999), 1165–1176 | DOI | MR | Zbl