Geodesic
Iteratively regularized Gauss--Newton method for operator equations with normally solvable derivative at the solution
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2016), pp. 3-11.

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

We study the iteratively regularized Gauss–Newton method in a Hilbert space for solving irregular nonlinear equations with smooth operators having normally solvable derivatives at the solution. We consider both a priori and a posteriori stopping criterions for the iterations and establish accuracy estimates for resulting approximations. In the case where the a priori stopping rule is used, the accuracy of approximations arises to be proportional to the error level in input data. The latter result generalizes well-known estimates of this kind obtained for linear equations with normally solvable operators.
Keywords: operator equation, irregular operator, Hilbert space, normally solvable operator, Gauss–Newton method, iterative regularization, stopping rule, accuracy estimate. 
@article{IVM_2016_8_a0,
     author = {A. B. Bakushinskii and M. Yu. Kokurin},
     title = {Iteratively regularized {Gauss--Newton} method for operator equations with normally solvable derivative at the solution},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--11},
     publisher = {mathdoc},
     number = {8},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_8_a0/}
}
TY  - JOUR
AU  - A. B. Bakushinskii
AU  - M. Yu. Kokurin
TI  - Iteratively regularized Gauss--Newton method for operator equations with normally solvable derivative at the solution
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2016
SP  - 3
EP  - 11
IS  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2016_8_a0/
LA  - ru
ID  - IVM_2016_8_a0
ER  - 
%0 Journal Article
%A A. B. Bakushinskii
%A M. Yu. Kokurin
%T Iteratively regularized Gauss--Newton method for operator equations with normally solvable derivative at the solution
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2016
%P 3-11
%N 8
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2016_8_a0/
%G ru
%F IVM_2016_8_a0
A. B. Bakushinskii; M. Yu. Kokurin. Iteratively regularized Gauss--Newton method for operator equations with normally solvable derivative at the solution. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2016), pp. 3-11. http://geodesic.mathdoc.fr/item/IVM_2016_8_a0/

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

[2] Izmailov A. F., Tretyakov A. A., $2$-regulyarnye resheniya nelineinykh zadach, Fizmatlit, M., 1999 | MR | Zbl

[3] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sib. nauchn. izd-vo, Novosibirsk, 2008

[4] Vainikko G. M., Veretennikov A. Yu., Iteratsionnye protsedury v nekorrektnykh zadachakh, Nauka, M., 1986 | MR

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

[6] Bakushinsky A., Smirnova A., “On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems”, Numer. Funct. Anal. Optim., 26:1 (2005), 35–48 | DOI | MR | Zbl

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

[8] Bakushinskii A. B., Kokurin M. Yu., Iteratsionnye metody resheniya neregulyarnykh uravnenii, LENAND, M., 2006

[9] Leonov A. S., “Mozhet li apriornaya otsenka tochnosti priblizhennogo resheniya nekorrektnoi zadachi byt sravnimoi s oshibkoi dannykh?”, Zhurn. vychisl. matem. i matem. fiz., 54:4 (2014), 562–568 | DOI | Zbl

[10] Kaltenbacher B., Neubauer A., Scherzer O., Iterative regularization methods for nonlinear ill-posed problems, Walter de Gruyter, Berlin, 2008 | MR

[11] Bakushinskii A. B., Kokurin M. Yu., Algoritmicheskii analiz neregulyarnykh operatornykh uravnenii, LENAND, M., 2012