Geodesic
Conditions for the qualified convergence of finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems in a Hilbert space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2019), pp. 46-61.

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

We consider finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems with selfadjoint operators and noise-free initial data in a Hilbert space. We refine the earlier author's results on the convergence rate of the methods under investigation. We establish the sufficient conditions and the necessary conditions, close to one another, for the qualified convergence of these methods in terms of the solution's sourcewise index. We prove that the considered methods cannot converge with the polynomial rate greater than the certain limit, except for the trivial case.
Keywords: ill-posed Cauchy problem, finite difference scheme, quasi-reversibility method, operator calculus, selfadjoint operator
Mots-clés : convergence rate, sourcewise representation, interpolation spaces. 
@article{IVM_2019_10_a5,
     author = {M. M. Kokurin},
     title = {Conditions for the qualified convergence of finite difference methods and the quasi-reversibility method for solving linear ill-posed {Cauchy} problems in a {Hilbert} space},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {46--61},
     publisher = {mathdoc},
     number = {10},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2019_10_a5/}
}
TY  - JOUR
AU  - M. M. Kokurin
TI  - Conditions for the qualified convergence of finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems in a Hilbert space
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2019
SP  - 46
EP  - 61
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2019_10_a5/
LA  - ru
ID  - IVM_2019_10_a5
ER  - 
%0 Journal Article
%A M. M. Kokurin
%T Conditions for the qualified convergence of finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems in a Hilbert space
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2019
%P 46-61
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2019_10_a5/
%G ru
%F IVM_2019_10_a5
M. M. Kokurin. Conditions for the qualified convergence of finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems in a Hilbert space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2019), pp. 46-61. http://geodesic.mathdoc.fr/item/IVM_2019_10_a5/

[1] Bakushinskii A. B., Kokurin M. Yu., Klyuchev V. V., “Ob otsenke skorosti skhodimosti i pogreshnosti raznostnykh metodov resheniya nekorrektnoi zadachi Koshi v banakhovom prostranstve”, Vychisl. metody i programmirov., 7 (2006), 163–171

[2] Ivanov V. K., Melnikova I. V., Filinkov A. I., Differentsialno-operatornye uravneniya i nekorrektnye zadachi, Fizmatlit, M., 1995

[3] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[4] Bakushinskii A. B., Kokurin M. M., Kokurin M. Yu., “Ob odnom klasse raznostnykh skhem resheniya nekorrektnoi zadachi Koshi v banakhovom prostranstve”, Zhurn. vychisl. matem. i matem. fiziki, 52:3 (2012), 483–498 | MR | Zbl

[5] Haase M., The functional calculus for sectorial operators, Birkhäuser, Basel, 2006 | MR | Zbl

[6] Riss F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979

[7] Trenogin V. A., Funktsionalnyi analiz, Fizmatlit, M., 2007

[8] Bakushinskii A. B., “Raznostnye metody resheniya nekorrektnykh zadach Koshi dlya evolyutsionnykh uravnenii v kompleksnom $B$-prostranstve”, Differents. uravneniya, 8:9 (1972), 1661–1668 | Zbl

[9] Kokurin M. M., “Ob optimizatsii otsenok skorosti skhodimosti nekotorykh klassov raznostnykh skhem resheniya nekorrektnoi zadachi Koshi”, Vychisl. metody i programmirov., 14 (2013), 58–76

[10] Kokurin M. M., “Neobkhodimye i dostatochnye usloviya stepennoi skhodimosti metoda kvaziobrascheniya i raznostnykh metodov resheniya nekorrektnoi zadachi Koshi v usloviyakh tochnykh dannykh”, Zhurn. vychisl. matem. i matem. fiziki, 55:12 (2015), 2027–2041 | Zbl

[11] Samarskii A. A., Vabischevich P. N., Chislennye metody resheniya obratnykh zadach matematicheskoi fiziki, Editorial URSS, M., 2004

[12] Tribel Kh., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980

[13] Lattes R., Lions Zh.-L., Metod kvaziobrascheniya i ego prilozheniya, Mir, M., 1970