Geodesic
Conditions of sourcewise representability and power estimates of convergence rate in Tikhonov's scheme for solving ill-posed extremal problems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2014), pp. 72-82.

Voir la notice de l'article provenant de la source Math-Net.Ru

We investigate rate of convergence estimates for approximations generated by Tikhonov's scheme for solving ill-posed optimization problems with general smooth functionals in a Hilbert space. Sourcewise representability conditions necessary and sufficient for convergence of approximations at the power rate are established. Sufficient conditions are related to estimates of a discrepancy by the objective functional while necessary ones are formulated for estimates by the argument. The cases are specified where sufficient and necessary conditions coincide in the main.
Keywords: ill-posed optimization problem, Hilbert space, Tikhonov's scheme, rate of convergence, sourcewise representability condition. 
@article{IVM_2014_7_a6,
     author = {M. Yu. Kokurin},
     title = {Conditions of sourcewise representability and power estimates of convergence rate in {Tikhonov's} scheme for solving ill-posed extremal problems},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {72--82},
     publisher = {mathdoc},
     number = {7},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2014_7_a6/}
}
TY  - JOUR
AU  - M. Yu. Kokurin
TI  - Conditions of sourcewise representability and power estimates of convergence rate in Tikhonov's scheme for solving ill-posed extremal problems
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2014
SP  - 72
EP  - 82
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2014_7_a6/
LA  - ru
ID  - IVM_2014_7_a6
ER  - 
%0 Journal Article
%A M. Yu. Kokurin
%T Conditions of sourcewise representability and power estimates of convergence rate in Tikhonov's scheme for solving ill-posed extremal problems
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2014
%P 72-82
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2014_7_a6/
%G ru
%F IVM_2014_7_a6
M. Yu. Kokurin. Conditions of sourcewise representability and power estimates of convergence rate in Tikhonov's scheme for solving ill-posed extremal problems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2014), pp. 72-82. http://geodesic.mathdoc.fr/item/IVM_2014_7_a6/

[1] Vasilev F. P., Metody resheniya ekstremalnykh zadach. Zadachi minimizatsii v funktsionalnykh prostranstvakh, regulyarizatsiya, approksimatsiya, Nauka, M., 1981 | MR

[2] Engl H. W., Hanke M., Neubauer A., Regularization of inverse problems, Kluwer, Dordrecht, 2000

[3] Bakushinskii A. B., Kokurin M. Yu., Iteratsionnye metody resheniya nekorrektnykh operatornykh uravnenii s gladkimi operatorami, Editorial URSS, M., 2002

[4] Kokurin M. Yu., “O vypuklosti funktsionala Tikhonova i iterativno regulyarizovannykh metodakh resheniya neregulyarnykh nelineinykh operatornykh uravnenii”, Zhurn. vychisl. matem. i matem. fiz., 50:4 (2010), 651–664 | MR | Zbl

[5] Kokurin M. Yu., “Ob organizatsii globalnogo poiska pri realizatsii skhemy Tikhonova”, Izv. vuzov. Matem., 2010, no. 12, 20–31 | MR | Zbl

[6] Kokurin M. Yu., “On sequential minimization of Tikhonov functionals in ill-posed problems with a priori information on solutions”, J. Inverse and Ill-posed Problems, 18:9 (2011), 1031–1050 | DOI | MR

[7] Tikhonov A. N., Leonov A. S., Yagola A. G., Nelineinye nekorrektnye zadachi, Nauka, Fizmatlit, M., 1995 | MR | Zbl

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

[9] Tautenhahn U., “On the method of Lavrentiev regularization for nonlinear ill-posed problems”, Inverse Problems, 18:1 (2002), 191–207 | DOI | MR | Zbl

[10] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl

[11] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov, Nauka, M., 1972 | MR