Abstract
For electrodynamic equations with permittivity specified by a symmetric matrix \(\varepsilon (x) = ({{\varepsilon }_{{ij}}}(x),i,j = 1,2,3)\), the inverse problem of determining this matrix from information on solutions of these equations is considered. It is assumed that the permittivity is a given positive constant \({{\varepsilon }_{0}} > 0\) outside a bounded domain \(\Omega \subset {{\mathbb{R}}^{3}}\), while, inside \(\Omega \), it is an anisotropic quantity such that the differences \({{\varepsilon }_{{ij}}}(x) - {{\varepsilon }_{0}}{{\delta }_{{ij}}} = :{{\tilde {\varepsilon }}_{{ij}}}(x),\)\(i,j = 1,2,3,\) are small. Here, \({{\delta }_{{ij}}}\) is the Kronecker delta. The inverse problem is studied in the linear approximation. The structure of the solution to a linearized direct problem for the electrodynamic equations is investigated, and it is proved that all elements of the matrix \(\tilde {\varepsilon }(x) = {{\tilde {\varepsilon }}_{{ij}}}(x),\;i,j = 1,2,3\), can be uniquely determined by special observation data. Moreover, the problem of recovering the diagonal components \({{\tilde {\varepsilon }}_{{ij}}}(x),\;i = 1,2,3,\) leads to a usual X-ray tomography problem, so these components can be efficiently computed. The recovery of the other components leads to a more complicated algorithmic procedure.
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REFERENCES
A. N. Tikhonov, “On the transient electric current in a homogeneous conducting half-space,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 10 (3), 213–231 (1946).
A. N. Tikhonov, “On the uniqueness of the solutions of the problems of electro-prospecting,” Dokl. Akad. Nauk SSSR 60 (5), 797–800 (1949).
A. N. Tikhonov, “Determination of the electrical characteristics of the deep strata of the Earth’s crust,” Dokl. Akad. Nauk SSSR 73 (2), 295–297 (1950).
A. N. Tikhonov, “Mathematical basis of the theory of electromagnetic soundings,” USSR Comput. Math. Math. Phys. 5 (3), 207–211 (1965).
L. Cagniard, “Basic theory of the magnetotelluric method of geophysical prospecting,” Geophysics 18 (3), 605–635 (1953).
V. G. Romanov, S. I. Kabanikhin, and T. P. Pukhnacheva, “On the theory of inverse problems of electrodynamics,” Dokl. Akad. Nauk SSSR 266 (5), 1070–1073 (1982).
V. G. Romanov, “On the uniqueness of the determination of the coefficients of Maxwell’s equations,” Nonclassical Problems for Equations of Mathematical Physics (Novosibirsk, Inst. Mat. Sib. Otd. Akad. Nauk SSSR, 1982), pp. 139–142 [in Russian].
V. G. Romanov and S. I. Kabanikhin, Inverse Problems in Geoelectrics (Nauka, Moscow, 1991) [in Russian].
S. He, S. I. Kabanikhin, V. G. Romanov, and S. Ström, “Analysis of the Green’s function approach to one-dimensional inverse problems: I. One parameter reconstruction,” J. Math. Phys. 34 (12), 5724–5746 (1993).
S. He, S. I. Kabanikhin, V. G. Romanov, and S. Ström, “Mathematical analysis of the Green’s function approach to the inverse problem: II. Simultaneous reconstruction,” J. Math. Phys. 35 (5), 2315–2335 (1994).
S. I. Kabanikhin and K. S. Abdiev, “Modeling the initial stage of transient electric current and using the results in the problem of determining the conductivity tensor,” Well-posedness of Inverse Problems of Mathematical Physics (Vychisl. Tsentr Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1982), pp. 85–94 [in Russian].
V. G. Romanov, “The structure of the fundamental solution of the Cauchy problem for the system of Maxwell’s equations,” Differ. Uravn. 22 (9), 1577–1587 (1986).
V. G. Romanov, “An inverse problem of electrodynamics,” Dokl. Math. 66 (2), 200–205 (2002).
V. G. Romanov, “Stability of the determination of the electrical conductivity in electrodynamic equations,” Dokl. Math. 67 (2), 167–171 (2003).
V. G. Romanov, “A stability estimate for a solution to a two-dimensional inverse problem of electrodynamics,” Sib. Math. J. 44 (4), 659–670 (2003).
V. G. Romanov, “A stability estimate for a solution to a three-dimensional inverse problem for the Maxwell equations,” Sib. Math. J. 45 (6), 1098–1112 (2004).
V. G. Romanov, “A stability estimate of the solution to the problem of determining dielectric permittivity and electric conductivity,” Dokl. Math. 71 (1), 154–159 (2005).
V. G. Romanov, “A stability estimate for a solution to an inverse problem of electrodynamics,” Sib. Math. J. 52 (4), 682–695 (2011).
V. G. Romanov, “Stability estimate of a solution to the problem of kernel determination in integrodifferential equations of electrodynamics,” Dokl. Math. 84 (1), 518–521 (2011).
V. G. Romanov, “The problem of determining the kernel of electrodynamics equations for dispersion media,” Dokl. Math. 84 (2), 613–616 (2011).
V. G. Romanov, “The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field,” Sib. Math. J. 58 (4), 711–717 (2017).
V. G. Romanov, “Problem of determining the permittivity in the stationary system of Maxwell equations,” Dokl. Math. 95 (3), 230–234 (2017).
V. G. Romanov, “Determination of permittivity from the modulus of the electric strength of a high-frequency electromagnetic field,” Dokl. Math. 99 (1), 44–47 (2019).
V. G. Romanov, “Uniqueness of the determination of dielectric permittivity and magnetic permeability in an anisotropic one-dimensional inhomogeneous medium,” Differ. Uravn. 20 (2), 325–332 (1984).
V. G. Romanov and M. G. Savin, “The problem of determining the conductivity tensor in a depth-inhomogeneous anisotropic medium,” Izv. Akad. Nauk SSSR, Ser. Fiz. Zemli, No. 2, 84–92 (1984).
V. G. Romanov and M. G. Savin, “Determination of the conductivity tensor in an anisotropic three-dimensional inhomogeneous medium: Linear approximation,” Izv. Akad. Nauk SSSR, Ser. Fiz. Zemli, No. 5, 63–72 (1984).
J. Radon, “Über die Bestimmung von Funktionen durch ihre integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Sachsische Akad. Wiss. Leipzig 29, 262–277 (1917).
S. Helgason, The Radon Transform (Birkhäuser, Boston, 1980).
S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
F. Natterer, The Mathematics of Computerized Tomography (SIAM, Philadelphia, PA, 2001).
Ju. E. Anikonov and V. G. Romanov, “On uniqueness of determination of a form of first degree by its integrals along geodesics,” J. Inv. Ill-Posed Probl. 5 (6), 487–490 (1997).
Funding
This work was supported by the comprehensive basic research program II.1 of the Siberian Branch of the Russian Academy of Sciences, project no. 0314-2018-0010.
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Translated by I. Ruzanova
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Romanov, V.G. Inverse Problem of Electrodynamics for Anisotropic Medium: Linear Approximation. Comput. Math. and Math. Phys. 60, 1037–1044 (2020). https://doi.org/10.1134/S0965542520060081
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DOI: https://doi.org/10.1134/S0965542520060081