Geodesic
Approximation of nonsmooth solutions of a retrospective problem for an advection-diffusion model
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 2, pp. 281-290 Cet article a éte moissonné depuis la source Math-Net.Ru

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A retrospective problem, which consists in recovering an a priori unknown initial state of a dynamical system from its known final state, is investigated. The time evolution of the system is described by a nonlinear boundary value problem for the inhomogeneous Burgers equation. This problem, as well as other similar problems, is ill- posed. We propose to solve the problem by Tikhonov's variational method, which consists in minimizing some suitable residual functional on the set of admissible solutions of the problem. The case of a discontinuous solutions is covered by employing stabilizers with the norm of the Sobolev space $W^\gamma_p([0,l])$ with fractional derivatives. For solving the extremal problems, iterative methods are proposed and justified, which reduce the initial unstable problem to a series of well-posed problems. A numerical investigation of the effectiveness of various stabilizers is carried out and the results of numerical calculations are presented.
Keywords: dynamical system, Burgers equation, inverse retrospective problem, Tikhonov's regularization method, classical variation, gradient method, subgradient. 
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I. A. Tsepelev. Approximation of nonsmooth solutions of a retrospective problem for an advection-diffusion model. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 2, pp. 281-290. http://geodesic.mathdoc.fr/item/TIMM_2012_18_2_a26/

[1] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979, 288 pp. | MR

[2] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978, 206 pp. | MR

[3] Lavrentev M. M., O nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki, SO AN SSSR, Novosibirsk, 1962, 92 pp. | MR

[4] Vasin V. V., “Approksimatsiya negladkikh reshenii lineinykh nekorrektnykh zadach”, Tr. In-ta matematiki i mekhaniki UrO RAN, 12, no. 1, 2006, 64–77 | MR | Zbl

[5] Bek Dzh., Blekuell B., Sent-Kler Ch. (ml.), Nekotorye obratnye zadachi teploprovodnosti, Mir, M., 1989, 312 pp.

[6] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Gos. izd-vo fiz.-mat. lit-ry, M., 1961, 204 pp.

[7] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR

[8] Temam R., Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981, 408 pp. | MR | Zbl

[9] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972, 587 pp. | MR | Zbl

[10] Korotkii A. I., Tsepelev I. A., “Solution of a Retrospective Inverse Problem for One Nonlinear Evolutionary Model”, Proc. Steklov Inst. Math., 2003, S80–S94 | MR | Zbl

[11] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp. | MR | Zbl

[12] Wang Y., Hutter K., “Comparisons of numerical methods with respect to convectively dominated problem”, Int. J. Numer. Meth. Fluid., 37 (2001), 721–745 | DOI | Zbl

[13] Anderson D., Tannekhill Dzh., Pletcher R., Vychislitelnaya gidromekhanika i teploobmen, v. 1, Mir, M., 1990, 384 pp. | Zbl

[14] Vasilev F. P., Metody optimizatsii, Faktorial Press, M., 2002, 824 pp.

[15] Sea Zh., Optimizatsiya. Teoriya i algoritmy, Mir, M., 1973, 244 pp.

[16] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 479 pp. | MR | Zbl

[17] Samarskii A. A., Gulin A. V., Chislennye metody, Nauka, M., 1989, 432 pp. | MR

[18] Samarskii A. A., Vabischevich P. N., Vasilev V. I., “Iteratsionnoe reshenie retrospektivnoi obratnoi zadachi teploprovodnosti”, Mat. modelirovanie, 9:5 (1997), 119–127 | MR | Zbl

[19] Samarskii A. A., “O regulyarizatsii raznostnykh skhem”, Zhurn. vychisl. matematiki i mat. fiziki, 7:1 (1967), 62–93 | MR | Zbl

[20] Samarskii A. A., Vabischevin P. N., “Nelineinye monotonnye skhemy dlya uravneniya perenosa”, Dokl. RAN, 361:1 (1998), 21–23 | MR

[21] OpenFOAM User guide, URL: , (data obrascheniya: 10.08.2011) http://foam.sourceforge.net/docs/Guides-a4/UserGuide.pdf

[22] OpenFOAM Programmers Guide, URL: , (data obrascheniya: 10.08.2011) http://foam.sourceforge.net/docs/Guides-a4/ProgrammersGuide.pdf