Geodesic
Iteratively regularized methods for irregular nonlinear operator equations with a normally solvable derivative at the solution
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 9, pp. 1543-1555 
M. Yu. Kokurin. Iteratively regularized methods for irregular nonlinear operator equations with a normally solvable derivative at the solution. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 9, pp. 1543-1555. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_9_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A group of iteratively regularized methods of Gauss–Newton type for solving irregular nonlinear equations with smooth operators in a Hilbert space under the condition of normal solvability of the derivative of the operator at the solution is considered. A priori and a posteriori methods for termination of iterations are studied, and estimates of the accuracy of approximations obtained are found. It is shown that, in the case of a priori termination, the accuracy of the approximation is proportional to the error in the input data. Under certain additional conditions, the same estimate is established for a posterior termination from the residual principle. These results generalize known similar estimates for linear equations with a normally solvable operator.

[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

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

[4] Kokurin M. Yu., “O vypuklosti funktsionala Tikhonova i iterativno regulyarizovannykh metodakh resheniya neregulyarnykh nelineinykh operatornykh uravnenii”, Zh. 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. vyssh. uchebn. zavedenii. Matematika, 2010, no. 12, 20–31

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

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

[8] 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

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

[10] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Fizmatlit, M., 2005

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

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

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