Geodesic
On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 96-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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A complete discretization scheme for an ill-posed Cauchy problem for abstract first-order linear differential equations with sectorial operators in a Banach space is validated. The scheme combines the time semidiscretization of the equations and a finite-dimensional approximation of the spaces and operators. Regularization properties of the scheme are established. Error estimates are obtained in the case of approximate initial data under various a priori assumptions concerning the solution.
Keywords: Cauchy problem, ill-posed problem, discretization, regularization, error estimate. 
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A. B. Bakushinskii; M. M. Kokurin; M. Yu. Kokurin. On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 96-108. http://geodesic.mathdoc.fr/item/TIMM_2012_18_1_a7/

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

[2] Bakushinsky A. B., Kokurin M. Yu., Kokurin M. M., “On a class of finite difference methods for ill-posed Cauchy problems with noisy data”, J. Inverse Ill-Posed Probl., 18:9 (2011), 959–977 | DOI | MR

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

[4] Bakaev N. Yu., Bakushinskii A. B., “K teorii priblizhennykh metodov resheniya nekorrektnoi zadachi Koshi”, Dokl. AN SSSR, 312:4 (1990), 777–782 | MR | Zbl

[5] Bakushinsky A. B., Kokurin M. Yu., Paymerov S. K., “On error estimates of difference solution methods for ill-posed Cauchy problems in a Hilbert space”, J. Inverse Ill-Posed Probl., 16:6 (2008), 553–565 | DOI | MR | Zbl

[6] Trenogin V. A., Funktsionalnyi analiz, Fizmatlit, M., 2007, 488 pp.

[7] Bukhgeim A. L., Vvedenie v teoriyu obratnykh zadach, Nauka, Cib. otd-nie, Novosibirsk, 1988, 184 pp. | MR

[8] Piskarëv S. I., “Otsenki skorosti skhodimosti pri reshenii nekorrektnykh zadach dlya evolyutsionnykh uravnenii”, Izv. AN SSSR. Ser. matem., 51:3 (1987), 676–687 | MR | Zbl

[9] Samarskii A. A., Vabischevich P. N., Chislennye metody resheniya obratnykh zadach matematicheskoi fiziki, Editorial URSS, M., 2004, 480 pp.

[10] Bakhvalov N. C., Zhidkov N. P., Kobelkov G. M., Chislennye metody, BINOM. Laboratoriya znanii, M., 2007, 636 pp.

[11] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967, 464 pp. | MR

[12] Ivanov V. K., Melnikova I. V., Filinkov A. I., Differentsialno-operatornye uravneniya i nekorrektnye zadachi, Fizmatlit, M., 1995, 176 pp. | MR

[13] Stewart H. B., “Generation of analytic semigroups by strongly elliptic operators”, Trans. Amer. Math. Soc., 199 (1974), 141–162 | DOI | MR

[14] Piskarëv S. I., “Ob otsenkakh skorosti skhodimosti pri poludiskretizatsii evolyutsionnykh uravnenii”, Differents. uravneniya, 19:12 (1983), 2153–2159 | MR | Zbl

[15] Alibekov Kh. N., Sobolevskii P. E., Ob ustoichivosti i skhodimosti raznostnykh skhem vysokogo poryadka approksimatsii dlya parabolicheskikh uravnenii. II, Dep. v VINITI 18.10.76, No 3645–B76, Voronezhskii gos. un-t, 50 pp.