Far Eastern Mathematical Journal

To content of the issue

Numerical analysis of the inverse identification problem for the lower coefficient of an elliptic equation

E. A. Kalinina

2005, issue 1-2, Ñ. 57–70

In this paper the inverse problem of identification of a coefficient in the two-dimensional stationary equation of diffusion – reaction is considered. For the solution of this problem the numerical algorithm is developed which is based on the two-layer gradient algorithm. Theoretical aspects and the convergence of the algorithm are discussed. The results of numerical experiments are analyzed in details.

elliptic equation, inverse extremal problem, coefficient identification, gradient algorithm, pollutant transfer

Download the article (PDF-file)


[1] G. V. Alekseev, “Obratnye e'kstremal'nye zadachi dlya stacionarnyx uravnenij teplomassoperenosa”, Dokl. RAN, 375:3 (2000), 315–319.
[2] C. V. Alekseev, E. A. Adomavichus, “Theoretical analysis of inverse extremal problems of admixture diffusion in viscous fluids”, J. Inv. Ill-Posed Problems, 9:5 (2001), 435–468.
[3] G. V. Alekseev, E'. A. Adomavichyus, “Issledovanie obratnyx e'kstremal'nyx zadach dlya nelinejnyx stacionarnyx uravnenij perenosa veshhestva”, Dal'nevost. mat. zhurn., 3:1 (2002), 79–92.
[4] G. V. Alekseev, “Obratnye e'kstremal'nye zadachi dlya stacionarnyx uravnenij teorii massoperenosa”, Zh. vychisl. matem. i matem. fiz., 42:3 (2002), 380–394.
[5] G. V. Alekseev, “Razreshimost' obratnyx e'kstremal'nyx zadach dlya stacionarnyx uravnenij teplomassoperenosa”, Sib. mat. zhurn., 42:5 (2001), 971–991.
[6] A. N. Tixonov, V. Ya. Arsenin, Metody resheniya nekorrektnyx zadach, Nauka, M., 1979.
[7] A. G. Fatullayev, “Determination of unknown coefficient in nonlinear diffusion equation”, Nonlinear Analysis, 44 (2001), 337–344.
[8] A. G. Fatullayev, “Numerical procedure for the determination of an unknown coefficients in parabolic equations”, Comp. Phys. Comm., 144 (2002), 29–33.
[9] K. Ito, K. Kunisch, “Estimation of the convection coefficient in elliptic equations”, Inverse Problems, 13 (1997), 995–1013.
[10] A. Shidfar, K. Tavakoli, “An inverse heat conduction problem”, South. Asian Bull. Math., 26 (2002), 503–507.
[11] M. Dehghan, “Finding a control parameter in one-dimensional parabolic equations”, Appl. Math. Comp., 135 (2003), 491–503.
[12] A. A. Samarskij, P. N. Vabishhevich, Chislennye metody resheniya obratnyx zadach matematicheskoj fiziki, Moskva, 2004, 480 s.
[13] B. Lowe, W. Rundell, “The determination of a coefficient in an elliptic equation from average flux data”, J. Comput. Math., 70 (1996), 173–187.
[14] D. A. Tereshko, “Chislennoe reshenie zadach identifikacii parametrov primesi dlya stacionarnyx uravnenij massoperenosa”, Vych. texn., 9:4, Spec. vyp. (2004), 92–98.
[15] A. Capatina, R. Stavre, “Numerical analysis of a control problem in heat conducting Navier – Stokes fluid”, Int. J. Eng. Sci., 34:13 (1996), 1467–1476.
[16] A. D. Ioffe, V. M. Tixomirov, Teoriya e'kstremal'nyx zadach, Nauka, M., 1974, 480 s.
[17] V. A. Trenogin, Funkcional'nyj analiz, Nauka, M., 1980, 496 s.

To content of the issue