Inverse Diffraction Problem of Determining the Properties of a Plasma Object
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INVERSE DIFFRACTION PROBLEM OF DETERMINING THE PROPERTIES OF A PLASMA OBJECT
UDC 532.59
². Ò. Selezov
Abstract. We consider the ill-posed problem for a plasma inhomogeneity, which is diagnosed by the reflected diffraction field, i.e., backscattering from the irradiated object. In this case, there is no need for a more accurate renewal of the object under study based on pseudo-inversion. An analytic solution of the inverse problem for a cylindrical plasma is constructed based on generalization of solutions for spherical plasma. The possibilities of plasma diagnostics from measured values are shown. Keywords: inverse problem, pseudoinversion, wave diffraction, backscattering, plasma diagnostics. INTRODUCTION The general problem of constructing solutions to inverse problems as ill-posed ones was always the focus of attention of contributors. The studies in solving inverse problems as ill-posed ones were developed by A. N. Tikhonov [1]. The Hadamard classical direct problem for a partial differential equation is formulated so that the solution exists, is unique, and is stable with respect to small variations of coefficients. In case of ill-posed problems, the solution is not stable with respect to small variations of data. Even insignificant errors (data variations) strongly increase; therefore, approximate solutions [2, 3] or solutions on the basis of pseudoinversion [4, 5] can only be obtained in most cases. Note that inverse problems occur in many areas of physics. Typical ones consist in renewal of certain functions from the measured data, for example, finding the field near a diffracting object or its form in acoustic scattering, establishing the form of scatterer from measurements of electromagnetic scattering, obtaining information about electronic density of plasma from measurements in exterior domain, etc. In these cases, there is no need in a more accurate renewal of the object of study [6]. The diffracting electromagnetic field from exterior emitters–receivers should not strongly influence the plasma density structure, and this can only be attained for very short wavelengths, in geometrical optics approximation. Solution of the problem of diffraction in geometrical optics approximation based on exact statement is known. Noteworthy is that plasma in a toroidal system with respect to control was studied at the Institute of Cybernetics, AS UkrSSR [7]. The problem of plasma diagnostics in a tokamak was analyzed in [8], where solution methods are presented for the inverse problem for the Schrodinger equation. The plasma diagnostics problem was permanently considered and is being considered at the Swiss Plasma Center (SPC) [9], where plasma diagnostics in physical devices and in a tokamak is investigated. The main concepts of various plasma diagnostics methods are presented in [10–12]. In case of weak electric conductivity for small magnetic Reynolds numbers (R m > 1) , the problem would be more simple [13]. However, in the present paper, we will consider the problem in the Institute of Hydromechanics, National
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