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Vasil’ev, V. I.; Kardashevskiĭ, A. M.

Iterative solution of the retrospective inverse heat conduction problem using the Poisson integral. (Russian. English summary) Zbl 1543.65220

Sib. Èlektron. Mat. Izv. 20, No. 1, 72-85 (2023).
Summary: This paper considers the inverse problem of identification of the finite initial condition of the Cauchy problem for the homogeneous heat conduction equation using the first kind linear Fredholm integral equation. Its discretization is carried out with the help of the quadrature rectangular formula. For the numerical realization of the obtained system of linear algebraic equations with almost complete, symmetric, positively determined, ill-conditioned matrix it is proposed to use the method of conjugate gradients. Examples of reconstruction of smooth, nonsmooth and discontinuous initial conditions in one- and two-dimensional cases, including the introduction of “noise”, characteristic of redefinition conditions of inverse problems, are given.

MSC:

65R32 Numerical methods for inverse problems for integral equations
65D32 Numerical quadrature and cubature formulas
80A23 Inverse problems in thermodynamics and heat transfer

Keywords:

retrospective inverse heat conduction problem; Poisson integral; first-kind Fredholm integral equation; system of linear equations with ill-conditioned matrix; method of conjugate gradients
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References:

[1] R. Lattès, J.-I. Lions, The method of quasi-reversibility. Applications to partial differential equations, Elsevier, New York, 1969 · Zbl 1220.65002
[2] V.K. Ivanov, V.V. Vasin, V.P. Tanana, Theory of linear ill-posed problems and its applications, Nauka, M., 1978 · Zbl 0489.65035
[3] A.N. Tikhonov, V.Ya. Arsenin, Methods for the solution of ill-posed problems, Nauka, M., 1979 · Zbl 0499.65030
[4] A.B. Vasil’eva, N.A. Tikhonov, Integral equations, Textbook, Izdatel’stvo Moskovskogo Universiteta, M., 1989 · Zbl 0687.45001
[5] A.N. Tikhonov, A.V. Goncharskij, V.V. Stepanov, A.G. Yagola, Numerical methods for the solution of ill-posed problems, Nauka, M., 1990 · Zbl 0712.65042
[6] A.M. Denisov, Vvedeniye v teoriyu obratnykh zadach, Izdatel’stvo Moskovskogo Universiteta, M., 1994
[7] M.M. Lavrent’ev, V.G. Romanov, S.T. Shishatskij, Ill-posed problems of mathematical physics and analysis, Nauka, M., 1980 · Zbl 0476.35001
[8] V.G. Romanov, Inverse problems of mathematical physics, Nauka, M., 1984 · Zbl 0576.35001
[9] A.A. Samarskii, P.N. Vabishchevich, Chislennyye metody resheniya obratnykh zadach matematicheskoy fiziki, Izdatel’stvo LKI, M., 2009
[10] S.I. Kabanikhin, Obratnyye i nekorrektnyye zadachi, Izdatel’stvo SO RAN, Novosibirsk, 2018
[11] A.I. Prilepko, D.G. Orlovsky, I.A. Vasin, Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, 2000 · Zbl 0947.35173
[12] V. Isakov, Inverse problems for partial differential equations, Springer, New York, 2006 · Zbl 1092.35001
[13] S.I. Kabanikhin, “Inverse problems of natural science”, Comput. Math. Math. Phys., 60:6 (2020), 911-914 · Zbl 1451.65248 · doi:10.1134/S0965542520060044
[14] S.I. Kabanikhin, G.B. Bakanov, “The optimizational method for solving the discrete inverse problem for hyperbolic equation”, J. Inverse Ill-Posed Probl., 4:6 (1996), 513-530 · Zbl 0909.35146 · doi:10.1515/jiip.1996.4.6.513
[15] S.I. Kabanikhin, G.B. Bakanov, “The optimization method for solving a discrete inverse problem for a hyperbolic equation”, Dokl. Math., 58:1 (1998), 32-35 · Zbl 0960.65101
[16] A.L. Karchevsky, “Finite-difference coefficient inverse problem and properties of the misfit functional”, J. Inverse Ill-Posed Probl., 6:5 (1998), 431-452 · Zbl 0920.35169 · doi:10.1515/jiip.1998.6.5.431
[17] A.A. Samarskii, P.N. Vabishchevich, V.I. Vasil’ev, “Iterative solution of a retrospective inverse problem of heat conduction”, Mat. Model., 9:5 (1997), 119-127 · Zbl 1071.80505
[18] D. Lesnic, L. Elliott, D. Ingham, “An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation”, Inverse Problems in Engineering, 6:4 (1998), 255-279 · doi:10.1080/174159798088027679
[19] A.I. Korotkii, I.A. Tsepelev, “Solution of a retrospective inverse problem for a nonlinear evolutionary model”, Proceedings of the Steklov Institute of Mathematics, 2003, Suppl. 2, S80-S94 · Zbl 1123.35356
[20] N.S. Mera, “The method of fundamental solutions for the backward heat conduction problem”, Inverse Probl. Sci. Eng., 13:1 (2005), 65-78 · Zbl 1194.80107 · doi:10.1080/10682760410001710141
[21] B.T. Johansson, D. Lesnic, “A procedure for determining a spacewise dependent heat source and the initial temperature”, Appl. Anal., 87:3 (2008), 265-276 · Zbl 1133.35436 · doi:10.1080/00036810701858193
[22] A. Ismail-Zadeh, A. Korotkii, G. Schubert, I. Tsepelev, “Numerical techniques for solving the inverse retrospective problem of thermal evolution of the Earth interior”, Computers & Structures, 87:11-12 (2009), 802-811 · doi:10.1016/j.compstruc.2009.01.005
[23] Zhenyu Zhao, Zehong Meng, “A modified Tikhonov regularization method for a backward heat equation”, Inverse Probl. Sci. Eng., 19:8 (2011), 1175-1182 · Zbl 1244.65136 · doi:10.1080/17415977.2011.605885
[24] V.I. Vasil’ev, A.M. Kardashevsky, P.V. Sivtsev, “Computational experiment on the numerical solution of some inverse problems of mathematical physics”, IOP Conf. Series: Mater. Sci. Eng., 158 (2016), 012093 · doi:10.1088/1757-899X/158/1/012093
[25] V.I. Vasil’ev, A.M. Kardashevsky, “Iterative solution of the retrospective inverse problem for a parabolic equation using the conjugate gradient method”, LNCS, 10187, Springer, Cham, 2017, 698-705 · Zbl 1368.65177
[26] V.P.. Tanana, A.I. Sidikova, “On an estimate for the error of an approximate solution caused by the discretization of an integral equation of the first kind”, Proc. Steklov Inst. Math., 2017, suppl. 1, S217-S224
[27] V.I. Vasil’ev, A.M. Kardashevskii, “Numerical solution of the retrospective inverse problem of heat conduction with the help of the Poisson integral”, J. Appl. Ind. Math., 12:3 (2018), 577-586 · Zbl 1413.65356 · doi:10.1134/S1990478918030158
[28] A. Savitzky, M.J.E. Golay, “Smoothing and differentiation of data by simplified least squares procedures”, Anal. Chem., 36:8 (1964), 1627-1639 · doi:10.1021/ac60214a047
[29] B.G. Gowri, V. Hariharan, S. Thara, etc., “2D Image data approximation using Savitzky Golay filter - Smoothing and differencing”, 2013 International Mutli-Conference on Automation, Computing, Communication, Control and Compressed Sensing (Kottayam, India, 2013), 365-371
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