Abstract
An iterative gradient descent method is applied to solve an inverse coefficient heat conduction problem with overdetermined boundary conditions. Theoretical estimates are derived showing how the target functional varies with varying the coefficient. These estimates are used to construct an approximation for a target functional gradient. In numerical experiments, iteration convergence rates are compared for different descent parameters.
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Lavrentiev, M.M., Romanov, V.G., and Shishatskii, S.P., Nekorrekinye zadachi matematicheskoi fiziki i analiza (Incorrect Problems of Mathematical Physics and Analysis), Moscow: Nauka, 1980.
Alifanov, O.M., Artyukhin, E.A., and Rumyantsev, S.V., Ekstremal’nye metody resheniya nekorektnykh zadach (Extreme Methods for Solving Ill-Posed Problems), Moscow: Nauka, 1988.
Hao, D., Methods for Inverse Heat Conduction Problems, Peter Lang, 1998.
Isakov, V. and Kindermann, S., Identifications of the Diffusion Coefficient in a One-Dimensional Parabolic Equation, Inv. Prob., 2000, no. 16, pp. 665–680.
Lishang, J. and Youshan, T., Identifying the Volatility of Underlying Assets from Option Prices, Inv. Prob., 2001, no. 17, pp. 137–155.
Plotnikov, V.I., Uniqueness and Existence Theorems and A Priori Properties of Generalized Solutions, Dokl. Akad. Nauk, 1965, no. 165, pp. 1405–1407.
Hasanov, A., DuChateau, P., and Pektas, B., An Adjoint Problem Approach and Coarse-Fine Mesh Method for Identification of the Diffusion Coefficient in a Linear Parabolic Equation, J. Inv. Ill-Pos. Probl., 2006, vol. 14, no. 4, pp. 1–29.
Vasiliev, F.P., Chislennye metody resheniya ekstremal’nykh zadach (Numerical Methods for Solving Extreme Problems), Moscow: Nauka, 1988.
Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1984.
Samarskii, A.A., Vvedenie v teoriyu raznostnykh skhem (Introduction to the Theory of Difference Schemes), Moscow: Nauka, 1971.
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Original Russian Text © S.I. Kabanikhin, A. Hasanov, A.V. Penenko, 2008, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2008, Vol. 11, No. 1, pp. 41–54.
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Kabanikhin, S.I., Hasanov, A. & Penenko, A.V. A gradient descent method for solving an inverse coefficient heat conduction problem. Numer. Analys. Appl. 1, 34–45 (2008). https://doi.org/10.1134/S1995423908010047
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DOI: https://doi.org/10.1134/S1995423908010047