Geodesic
Determination of the diffusion coefficient
Matematičeskoe modelirovanie, Tome 27 (2015) no. 1, pp. 16-32.

Voir la notice de l'article provenant de la source Math-Net.Ru

The multilevel algorithm for the solution of the inverse problem for the diffusion equation by an optimization method using the Laguerre functions is considered. Numerical simulations are carried out for Maxwell's equations in 1-dimensional setting in diffusion approximations. Spatial distributions of conductivity of the medium are determined from a known solution in some point. A function of the Laguerre harmonics is minimized. Minimizations are performed by Newton and the conjugate gradient method. The influence of form and spectrum of the source of electromagnetic waves on the accuracy of solution of the inverse problem is investigated. The accuracy of the solution of the inverse problem when using multilevel algorithm and the conventional one-level algorithm are compared.
Mots-clés : diffusion coefficient, Laguerre method.
Keywords: Maxwell’s equations, multilevel algorithm 
@article{MM_2015_27_1_a1,
     author = {A. F. Mastryukov},
     title = {Determination of the diffusion coefficient},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {16--32},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2015_27_1_a1/}
}
TY  - JOUR
AU  - A. F. Mastryukov
TI  - Determination of the diffusion coefficient
JO  - Matematičeskoe modelirovanie
PY  - 2015
SP  - 16
EP  - 32
VL  - 27
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2015_27_1_a1/
LA  - ru
ID  - MM_2015_27_1_a1
ER  - 
%0 Journal Article
%A A. F. Mastryukov
%T Determination of the diffusion coefficient
%J Matematičeskoe modelirovanie
%D 2015
%P 16-32
%V 27
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2015_27_1_a1/
%G ru
%F MM_2015_27_1_a1
A. F. Mastryukov. Determination of the diffusion coefficient. Matematičeskoe modelirovanie, Tome 27 (2015) no. 1, pp. 16-32. http://geodesic.mathdoc.fr/item/MM_2015_27_1_a1/

[1] L. D. Landau, E. M. Lifshits, Elektrodinamika sploshnykh sred, Nauka, M., 1982, 620 pp.

[2] A. G. Tarkhov (ed.), Elektrorazvedka. Spravochnik geofizika, Nedra, M., 1980, 518 pp.

[3] J. Beck, B. Blackwell, C. St.-Clair, Inverse heat conduction ill-posed problem, J. Wiley Inc., NY., 1985, 312 pp.

[4] A. N. Tikhonov, V. Ia. Arsenin, Metody resheniia nekorrektnykh zadach, Nauka, M., 1979, 385 pp.

[5] R. P. Fedorenko, Priblizhennoe reshenie zadach optimalnogo upravleniia, Nauka, M., 1978, 391 pp.

[6] F. P. Vasilev, Chislennye metody resheniia ekstremalnykh zadach, Nauka, M., 1980, 518 pp.

[7] V. V. Plotkin, “Magnitotelluricheskoe zondirovanie plavno-neodnorodnoi anizotropnoi sredy”, Geologiia i geofizika, 53:8 (2012), 1078–1089

[8] W. Hu, A. Abubakar, T. M. Habashy, J. Liu, “Preconditioned non-linear conjugate gradient method for frequency domain full-waveform seismic inversion”, Geophysical Prospecting, 59 (2011), 477–491 | DOI

[9] G. A. Newton, D. L. Alumbaugh, “Three-dimensional magnetotelluric inversion using non-linear conjugate gradients”, Geophysical Journal International, 140:3 (2000), 410–424

[10] A. F. Mastriukov, “Reshenie obratnoi zadachi dlia uravneniia diffuzii na osnove spektralnogo preobrazovaniia Lagerra”, Matematicheskoe modelirovanie, 2007, no. 9, 15–26

[11] A. F. Mastriukov, B. G. Mikhailenko, “Reshenie obratnoi zadachi dlia volnovogo uravneniia na osnove spektralnogo preobrazovaniia Lagerra”, Geologiia i geofizika, 44:7 (2007), 747–754

[12] B. G. Mikhailenko, “Spectral Laguerre method for the approximate solution of time dependent problems”, Applied Mathematics Letters, 12 (1999), 105–110 | DOI

[13] M. Abramowitz, I. Stegun, Handbook of mathematical functions, National Bureau of Standards, 1964, 1046 pp.

[14] C. Bunks, F. M. Saleck, S. Zaleski, G. Chavent, “Multiscale seismic waveform inversion”, Geophysics, 60:5 (1995), 1457–1473 | DOI

[15] A. F. Mastriukov, “Vosstanovlenie dvumernoi provodimosti sredy optimizatsionnym metodom”, Geologiia i geofizika, 1997, no. 3, 686–692

[16] R. P. Fedorenko, “Relaksatsionnyi metod dlia resheniia ellipticheskikh uravnenii”, ZHVMiMF, 1961, no. 1, 992–927

[17] J. Demmel, Applied numerical linear algebra, SIAM, Philadelphia, 1997, 425 pp.

[18] M. I. Epov, V. L. Mironov, S. A. Komarov, K. V. Muzalevskii, “Rasprostranenie sverkhshirokopolosnogo elektromagnitnogo impulsa v neftenasyshchennoi srede”, Geologiia i geofizika, 2009, no. 1, 58–66