Geodesic
A posteriori accuracy estimations of solutions of ill-posed inverse problems and extra-optimal regularizing algorithms for their solution
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 1, pp. 83-100 
A. S. Leonov. A posteriori accuracy estimations of solutions of ill-posed inverse problems and extra-optimal regularizing algorithms for their solution. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 1, pp. 83-100. http://geodesic.mathdoc.fr/item/SJVM_2012_15_1_a6/
@article{SJVM_2012_15_1_a6,
     author = {A. S. Leonov},
     title = {A posteriori accuracy estimations of solutions of ill-posed inverse problems and extra-optimal regularizing algorithms for their solution},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {83--100},
     year = {2012},
     volume = {15},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2012_15_1_a6/}
}
TY  - JOUR
AU  - A. S. Leonov
TI  - A posteriori accuracy estimations of solutions of ill-posed inverse problems and extra-optimal regularizing algorithms for their solution
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2012
SP  - 83
EP  - 100
VL  - 15
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SJVM_2012_15_1_a6/
LA  - ru
ID  - SJVM_2012_15_1_a6
ER  - 
%0 Journal Article
%A A. S. Leonov
%T A posteriori accuracy estimations of solutions of ill-posed inverse problems and extra-optimal regularizing algorithms for their solution
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2012
%P 83-100
%V 15
%N 1
%U http://geodesic.mathdoc.fr/item/SJVM_2012_15_1_a6/
%G ru
%F SJVM_2012_15_1_a6

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

A new scheme of a posteriori accuracy estimation for approximate solutions of ill-posed inverse problems is presented along with an algorithm of calculating this estimation. A new notion of extra-optimal regularizing algorithm is introduced as a method for solving ill-posed inverse problems having optimal in order a posteriori accuracy estimation. Sufficient conditions of extra-optimality are formulated and an example of extra-optimal regularizing algorithm is given. The developed theory is illustrated by numerical experiments.

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

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

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

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

[5] Leonov A. S., Reshenie nekorrektno postavlennykh obratnykh zadach. Ocherk teorii, prakticheskie algoritmy i demonstratsii v MATLAB, URSS, M., 2009

[6] Tanana V. P., Metody resheniya operatornykh uravnenii, Nauka, M., 1981 | MR

[7] Tanana V. P., Rekant M. A., Yanchenko S. I., Optimizatsiya metodov resheniya operatornykh uravnenii, Izd-vo Uralskogo universiteta, Sverdlovsk, 1987 | MR | Zbl

[8] Vainikko G. M., Metody resheniya lineinykh nekorrektno postavlennykh zadach v gilbertovykh prostranstvakh, Izd-vo TGU, Tartu, 1982 | MR

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

[10] Engl H. W., Hanke M., Neubauer A., Regularization of Inverse Problems, Kluwer Academic Publ., Dordrecht, 1996 | MR | Zbl

[11] Dombrovskaya I. N., Ivanov V. K., “K teorii lineinykh uravnenii v abstraktnykh prostranstvakh”, Sib. mat. zhurnal, 6:3 (1965), 499–508

[12] Gaponenko Yu. L., Vinokurov V. A., “Aposteriornye otsenki resheniya nekorrektnykh obratnykh zadach”, DAN SSSR, 263:2 (1982), 277–280 | MR | Zbl

[13] Yagola A. G., Nikolaeva N. N., Titarenko V. N., “Otsenka pogreshnosti resheniya uravneniya Abelya na mnozhestvakh monotonnykh i vypuklykh funktsii”, Sib. zhurn. vychisl. matematiki, 6:2 (2003), 171–180 | Zbl

[14] Titarenko V. N., Yagola A. G., “The problems of linear and quadratic programming for ill-posed problems on some compact sets”, J. Inverse and Ill-posed Problems, 11:3 (2003), 311–328 | DOI | MR | Zbl

[15] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl

[16] Vasilev F. P., Metody optimizatsii, Faktorial Press, M., 2002

[17] Leonov A. S., “Ob aposteriornykh otsenkakh tochnosti resheniya lineinykh nekorrektno postavlennykh zadach i ekstraoptimalnykh regulyarizuyuschikh algoritmakh”, Vychisl. metody i programmirovanie, 11:1 (2010), 14–24

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

[19] Tikhonov A. N., Goncharskii A. V., Stepanov V. V., Yagola A. G., Chislennye metody resheniya nekorrektnykh zadach, Nauka, M., 1990 | MR | Zbl

[20] Leonov A. S., “Ob ustranenii nasyscheniya tochnosti regulyarizuyuschikh algoritmov”, Sib. zhurn. vychisl. matematiki, 11:2 (2008), 167–186 | Zbl

[21] Chavent G., Non Linear Least Squares For Inverse Problems. Scientific Computation, Springer, 2009 | Zbl