Geodesic
Regularizing algorithms for detecting discontinuities in ill-posed problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1362-1370 
A. L. Ageev; T. V. Antonova. Regularizing algorithms for detecting discontinuities in ill-posed problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1362-1370. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a2/
@article{ZVMMF_2008_48_8_a2,
     author = {A. L. Ageev and T. V. Antonova},
     title = {Regularizing algorithms for detecting discontinuities in ill-posed problems},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1362--1370},
     year = {2008},
     volume = {48},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a2/}
}
TY  - JOUR
AU  - A. L. Ageev
AU  - T. V. Antonova
TI  - Regularizing algorithms for detecting discontinuities in ill-posed problems
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 1362
EP  - 1370
VL  - 48
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a2/
LA  - ru
ID  - ZVMMF_2008_48_8_a2
ER  - 
%0 Journal Article
%A A. L. Ageev
%A T. V. Antonova
%T Regularizing algorithms for detecting discontinuities in ill-posed problems
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 1362-1370
%V 48
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a2/
%G ru
%F ZVMMF_2008_48_8_a2

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

The problem of detecting singularities (discontinuities of the first kind) of a noisy function in $L_2$ is considered. A wide class of regularizing algorithms that can detect discontinuities is constructed. New estimates of accuracy of determining the location of discontinuities are obtained and their optimality in terms of order with respect to the error level $\delta$ is proved for some classes of functions with isolated singularities. New upper bounds for the singularity separation threshold are obtained.

[1] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1974 | MR

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

[3] Vasin V. V., Ageev A. L., Ill-posed problems with a priori information, VSP, Utrecht, Netherlands, 1995 | MR | Zbl

[4] Ageev A. L., Antonova T. V., “On solution of nonlinear with respect to parameter equation of the first kind on the class of discontinuous functions”, J. Inverse and Ill-Posed Problems, 7:1 (1999), 1–16 | DOI | MR | Zbl

[5] Ageev A. L., Antonova T. V., Baux T. E. i dr., “Metod razdelyayuschikh funktsionalov pri rasshifrovke lokalnoi atomnoi struktury”, Matem. modelirovanie, 16:10 (2004), 81–92 | Zbl

[6] Ageev A. L., Korshunov M. E., “Obrabotka izobrazhenii s vydeleniem polozheniya ob'ektov”, Tr. Mezhdunar. seminara: Teoriya upravleniya i teoriya obobschennykh reshenii uravnenii Gamiltona–Yakobi (Ekaterinburg, 22–23 iyunya 2005 g.), v. 2, Ekaterinburg, 2006, 61–65

[7] Antonova T. V., “Vosstanovlenie funktsii s konechnym chislom razryvov I roda po zashumlennym dannym”, Izv. vuzov. Matem., 2001, no. 7, 65–68 | MR | Zbl

[8] Antonova T. V., “Recovery of function with finite number of discontinuities by noised data”, J. Inverse and Ill-Posed Problems, 10:2 (2002), 1–11 | MR

[9] Antonova T. B., “O reshenii uravnenii I roda na klassakh razryvnykh funktsii”, Probl. teor. i prikl. matem. Tr. XXXI Regionalnoi molodezhnoi konf., UrO RAN, Ekaterinburg, 2000, 30–31

[10] Antonova T. V., “O reshenii nelineinykh po parametru uravnenii I roda na klassakh obobschennykh funktsii”, Zh. vychisl. matem. i matem. fiz., 40:6 (2000), 819–831 | MR | Zbl

[11] Antonova T. V., Reshenie nelineinykh uravnenii I roda na klassakh funktsii s razryvami, Dep. v VINITI 17.10.00, No 2639-V00

[12] Antonova T. V., “Solving equations of the first kind on classes of functions with singularities”, Proc. Steklov Inst. Math., Suppl. 1, 2002, 145–189 | MR

[13] Malla S., Veivlety v obrabotke signalov, Mir, M., 2005

[14] Terebizh V. Yu., Vvedenie v statisticheskuyu teoriyu obratnykh zadach, Fizmatlit, M., 2005

[15] Kozlov V. P., “O razreshayuschei sposobnosti spektralnykh priborov. I. Postanovka zadachi i kriterii razresheniya”, Optika i spektroskopiya, 16:3 (1964), 501–506

[16] Kozlov V. P., “O razreshayuschei sposobnosti spektralnykh priborov. II. Obobschennaya razreshayuschaya sila spektralnogo pribora”, Optika i spektroskopiya, 17:2 (1964), 278–283