Geodesic
Numerical implementation of special regularizing algorithms for solving a~class of ill-posed problems with sourcewise represented solutions
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 269-280.

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

An important class of ill-posed problems, namely, inverse problems of continuation for the values of an abstract function, is under consideration. The class is associated with certain semigroups of operators. A priori, the problems of this class have sourcewise represented solutions with known powers. For solving these problems we apply special Tikhonov's regularizing algorithms of the generalized residual principle. The approximate solutions obtained by the use of the algorithms and modifications proposed are of optimal order of accuracy without regard to the positive sourcewise representation power. The algorithm is numerically implemented in an efficient way. An example of application of the algorithm is given. 
@article{SJVM_2001_4_3_a4,
     author = {A. S. Leonov},
     title = {Numerical implementation of special regularizing algorithms for solving a~class of ill-posed problems with sourcewise represented solutions},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {269--280},
     publisher = {mathdoc},
     volume = {4},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a4/}
}
TY  - JOUR
AU  - A. S. Leonov
TI  - Numerical implementation of special regularizing algorithms for solving a~class of ill-posed problems with sourcewise represented solutions
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2001
SP  - 269
EP  - 280
VL  - 4
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a4/
LA  - ru
ID  - SJVM_2001_4_3_a4
ER  - 
%0 Journal Article
%A A. S. Leonov
%T Numerical implementation of special regularizing algorithms for solving a~class of ill-posed problems with sourcewise represented solutions
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2001
%P 269-280
%V 4
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a4/
%G ru
%F SJVM_2001_4_3_a4
A. S. Leonov. Numerical implementation of special regularizing algorithms for solving a~class of ill-posed problems with sourcewise represented solutions. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 269-280. http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a4/

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

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

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

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

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

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

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

[8] Leonov A. S., Yagola A. G., “Adaptivnye regulyarizuyuschie algoritmy dlya resheniya nekorrektnykh zadach”, Vestn. Mosk. un-ta. Seriya 3. Fizika. Astronomiya, 1998, no. 2, 62–63 | MR

[9] Leonov A. S., Yagola A. G., “Special regularizing methods for ill-posed problems with sourcewise represented solutions”, Inverse Problems, 14 (1998), 1539–1550 | DOI | MR | Zbl

[10] Rise F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR

[11] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[12] Tikhonov A. N., Glasko V. B., “O priblizhennom reshenii integralnykh uravnenii Fredgolma I roda”, Zhurn. vychisl. matem. i mat. fiz., 4:3 (1964), 564–571 | Zbl

[13] Balakrishnan A. V., Prikladnoi funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl

[14] Morozov V. A., “O printsipe nevyazki pri reshenii operatornykh uravnenii metodom regulyarizatsii”, Zhurn. vychisl. matem. i mat. fiz., 8:2 (1968), 295–309 | Zbl

[15] Strakhov V. N., “O reshenii nekorrektnykh zadach magnito i gravimetrii, predstavlyaemykh integralnymi uravneniyami tipa svertki”, Izv. AN SSSR. Fizika Zemli, 4:5 (1967), 33–53

[16] Goncharskii A. V., Leonov A. S., Yagola A. G., “O reshenii dvumernykh uravnenii Fredgolma 1-go rola s yadrom, zavisyaschim ot raznosti argumentov”, Zhurn. vychisl. matem. i mat. fiz., 11:5 (1971), 1296–1301

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