Geodesic
Iterative methods of stochastic approximation for solving non-regular nonlinear operator equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 10, pp. 1637-1645 Cet article a éte moissonné depuis la source Math-Net.Ru

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Iterative methods for solving non-regular nonlinear operator equations in a Hilbert space under random noise are constructed and examined. The methods use the averaging of the input data. It is not assumed that the noise dispersion is known. An iteratively regularized method of order zero for equations with monotone operators and iteratively regularized methods of the Gauss–Newton type for equations with arbitrary smooth operators are used as the basic procedures. It is shown that the generated approximations converge in the mean-square sense to the desired solution or stabilize (again in the mean-square sense) in a small neighborhood of the solution. 
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A. B. Bakushinskii; M. Yu. Kokurin. Iterative methods of stochastic approximation for solving non-regular nonlinear operator equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 10, pp. 1637-1645. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_10_a1/

[1] Vazan M., Stokhasticheskaya approksimatsiya, Mir, M., 1972

[2] Nevelson M. B., Khasminskii R. Z., Stokhasticheskaya approksimatsiya i rekurrentnoe otsenivanie, Nauka, M., 1972 | MR

[3] Kushner H. J., Yin G. G., Stochastic approximation and recursive algorithms and applications, Springer, New York, 2003 | MR | Zbl

[4] Salov G. I., “Ob odnoi teoreme stokhasticheskoi approksimatsii v gilbertovom prostranstve i ee prilozheniyakh”, Teoriya veroyatnostei i ee primeneniya, 24:2 (1979), 407–413 | MR

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

[6] Fedotov A. M., Nekorrektnye zadachi so sluchainymi oshibkami v iskhodnykh dannykh, Nauka, Novosibirsk, 1990 | MR

[7] Kaipio J., Somersalo E., Statistical and computational inverse problems, Springer, New York, 2005 | MR | Zbl

[8] Bauer F., Hohage T., Munk A., “Iteratively regularized Gauss-Newton method for nonlinear inverse problems with random noise”, SIAM J. Numer. Analys., 47:3 (2009), 1827–1846 | DOI | MR | Zbl

[9] Tsypkin Ya. Z., Osnovy teorii obuchayuschikhsya sistem, Nauka, M., 1970 | MR

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

[11] Bakushinskii A. B., “O postroenii regulyarizuyuschego algoritma pri sluchainykh pomekhakh”, Dokl. AN SSSR, 189:2 (1969), 231–233 | MR | Zbl

[12] Bakushinskii A. B., Apartsin A. S., “Metody tipa stokhasticheskoi approksimatsii dlya resheniya lineinykh nekorrektnykh zadach”, Sibirskii matem. zhurnal, 16:1 (1975), 12–18 | MR

[13] Bruck R. E., “On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space”, J. Math. Anal. Appl., 61:1 (1977), 159–164 | DOI | MR | Zbl

[14] Nemirovskii A. S., Yudin D. B., Slozhnost zadach i effektivnost metodov optimizatsii, Nauka, M., 1979 | MR

[15] Polyak B. T., “Novyi metod tipa stokhasticheskoi approksimatsii”, Avtomatika i telemekhan., 1990, no. 7, 98–107

[16] Schwable R., “On Bather's stochastic approximation algorithm”, Kybernetika, 30:3 (1994), 301–306 | MR

[17] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985 | MR

[18] Kakhan Zh.-P., Sluchainye funktsionalnye ryady, Mir, M., 1973 | MR

[19] Kokurin M. Yu., Operatornaya regulyarizatsiya i issledovanie nelineinykh monotonnykh zadach, Izd-vo Mar. GU, Ioshkar-Ola, 1998

[20] Gan S., Qui D., “On the Hajek–Renyi inequality”, Wuhan Univ. J. Nat. Sci., 12:6 (2007), 971–974 | DOI | MR | Zbl

[21] Miao Y., “Hajek–Renyi inequality for dependent random variables in Hilbert space and applications”, Rev. Un. Mat. Argentina, 53:1 (2012), 101–112 | MR | Zbl

[22] Petrov V. V., Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987 | MR

[23] Bakushinskii A. B., Kokurin M. Yu., Kozlov A. I., Stabiliziruyuschiesya metody gradientnogo tipa dlya resheniya neregulyarnykh nelineinykh operatornykh uravnenii, Izdatelstvo LKI, M., 2007

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