Geodesic
Iterative method for the Dirichlet problem and its modifications
Matematičeskie zametki SVFU, Tome 24 (2017) no. 3, pp. 38-51 Cet article a éte moissonné depuis la source Math-Net.Ru

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A series of works of S. I. Kabanikhin's scientific school are devoted to study of the existence, uniqueness, and numerical methods for the inverse Dirichlet problem for the second-order hyperbolic equations. We consider a numerical solution to the non-classical Dirichlet problem and its modifications for the two-dimensional hyperbolic second-order equations. The method of iterative refinement of the missing initial condition is applied by means of an additional condition specified at the final time. Moreover, the direct problem is numerically realized at each iteration. The efficiency of the proposed computational algorithm is confirmed by calculations for two-dimensional model problems, including additional conditions with random errors.
Keywords: hyperbolic equation, inverse problem, Dirichlet problem, finite difference method, iterative method, conjugate gradients method, random error. 
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V. I. Vasiliev; A. M. Kardashevsky; V. V. Popov. Iterative method for the Dirichlet problem and its modifications. Matematičeskie zametki SVFU, Tome 24 (2017) no. 3, pp. 38-51. http://geodesic.mathdoc.fr/item/SVFU_2017_24_3_a3/

[1] Kabanikhin S. I., Inverse and ill-posed problems. Theory and applications, De Gruyter, Berlin, 2011 | MR

[2] Lavrent'ev M. M., Romanov V. G., Shishatskii S. P., Ill-posed problems of mathematical physics and analysis, Amer. Math. Soc., Providence, RI, 1986 | MR | Zbl

[3] Samarskii A. A., Vabishchevich P. N., Numerical methods for solving inverse problems of mathematical physics, De Gruyter, Berlin, 2007 | MR | Zbl

[4] Kabanikhin S. I., Bektemesov M. A., Nurseitov D. B., Krivorotko O. I., Alimova A. N., “An optimization method in the Dirichlet problem for the wave equation”, J. Inverse Ill-posed Probl., 20:2 (2012), 193–211 | DOI | MR | Zbl

[5] Kabanikhin S. I., Krivorotko O. I., “A numerical method for solving the Dirichlet problem for the wave equation”, J. Appl. Ind. Math., 7:2 (2013), 187–198 | DOI | MR | Zbl

[6] Samarskii A. A., Vabishchevich P. N., Vasil'ev V. I., “Iterative solution of a retrospective inverse problem of heat conduction”, Mat. Model., 9:5 (1997), 119–127 | MR | Zbl

[7] Vasil'ev V. I., Popov V. V., Eremeeva M. S., Kardashevsky A. M., “Iterative solution of a non classical problem for the equation of string vibrations”, Vestn. Mosk. Gos. Tekh. Univ. Im. N. Eh. Baumana, Ser. Estestv. Nauki, 2015, no. 3, 77–87

[8] Vabishchevich P. N., Vasil'ev V. I., “terative solution of the Dirichlet problem for hyperbolic equations”, Setochnye Metody dlya Kraevykh Zadach i Prilozheniya, Mat. X Mezhdunar. Konf., Izd-vo Kazansk. Univ., Kazan, 2014, 162–166

[9] Vasil'ev V. I., Kardashevsky A. M., “Iterative solutions to some inverse problems for second-order hyperbolic equations”, Aktualnye Problemy Vychislitelnoi i Prikladnoi Matematiki, Tr. Mezhdunar. Konf., Abvei, Novosibirsk, 2015, 150–156

[10] Samarskii A. A., The theory of difference schemes, Marcel Dekker, New York; Basel, 2001 | MR | Zbl

[11] Saad Yu., Iterative methods for sparse linear systems, 2nd ed., SIAM, Philadelphia, PA, 2003 | MR | Zbl