Abstract
A multilevel algorithm for solving the inverse problem for the diffusion equation by the optimization method using Laguerre functions is considered. Numerical calculations are carried out for Maxwell’s equations in a one-dimensional formulation in the diffusion approximations. Using the known solution, we find the medium’s conductivity distribution at some point of space. The function of the Laguerre harmonics is minimized by Newton’s method and the conjugate gradients technique. We investigate the influence of the form of the source of the electromagnetic waves and its spectrum on the solution’s accuracy for the inverse problem. The accuracy of this solution obtained by a multi-level algorithm and using conventional single-level algorithms is compared.
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References
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Nauka, Moscow, 1982; Pergamon, New York, 1984).
Electrical Prospecting. Geophysicist’s Manual, Ed. by G. Tarkhov (Nedra, Moscow, 1980) [in Russian].
J. Beck, B. Blackwell, and C. St.-Clair, Inverse Heat Conduction Ill-Posed Problem (Wiley, New York, 1985; Mir, Moscow, 1989).
A. N. Tikhonov and V. Ya. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977; Nauka, Moscow, 1979).
R. P. Fedorenko, Approximate Solution of Optimal Control Problems (Nauka, Moscow, 1978) [in Russian].
F. P. Vasil’ev, Numerical Methods for Solving Extremal Problems (Nauka, Moscow, 1980), p. 518 [in Russian].
V. V. Plotkin, “Inversion of heterogeneous anisotropic magnetotelluric responces,” Russ. Geol. Geophys. 53, 829 (2012).
A. Hu, T. M. Abubakar, and J. Liu. Habashy, “Preconditioned non-linear conjugate gradient method for frequency domain full-waveform seismic inversion,” Geophys. Prospect. 59, 477–491 (2011).
G. A. Newton and D. L. Alumbaugh, “Three-dimensional magnetotelluric inversion using non-linear conjugate gradients,” Geophys. J. Int. 140, 410–424 (2000).
A. F. Mastryukov, “Inverse solution to the diffusion equation using the Laguerre spectral transform,” Mat. Model., No. 9, 15–26 (2007).
A. F. Mastryukov and B. G. Mikhailenko, “Inverse solution to the wave equation using the Laguerre transform,” Russ. Geol. Geophys. 44, 575 (2007).
B. G. Mikhailenko, “Spectral Laguerre method for the approximate solution of time dependent problems,” Appl. Math. Lett. 12, 105–110 (1999).
Handbook of Mathematical Functions, Ed. by M. Abramowitz and I. Stegun (Nation. Bureau of Standards, New York, 1964; Moscow, Nauka, 1979).
C. Bunks, F. M. Saleck, S. Zaleski, and G. Chavent, “Multiscale seismic waveform inversion,” Geophys. 60, 1457–1473 (1995).
A. F. Mastryukov, “Recovery of two-dimensional conductivity of medium by optimization method,” Geol. Geofiz., No. 3, 686–692 (1997).
R. P. Fedorenko, “A relaxation method for solving elliptic difference equations,” Zh. Vychisl. Mat. Mat. Fiz. 1, 922–927 (1961).
J. Demmel, Applied Numerical Linear Algebra (SIAM, Philadelphia, 1997).
M. I. Epov, V. L. Mironov, S. A. Komarov, and K. V. Muzalevskii, “Ultrabroadband electromagnetic wave propagation in hydrocarbon reservoirs in the presence of an oil-water interface,” Russ. Geol. Geophys., No. 1, 46 (2009).
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Original Russian Text © A.F. Mastryukov, 2015, published in Matematicheskoe Modelirovanie, 2015, Vol. 27, No. 1, pp. 16–32.
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Mastryukov, A.F. Determination of the diffusion coefficient. Math Models Comput Simul 7, 349–359 (2015). https://doi.org/10.1134/S2070048215040067
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DOI: https://doi.org/10.1134/S2070048215040067