Numerical Complex Analysis Method for Parameters Identification of Anisotropic Media Using Applied Quasipotential Tomographic Data. Part 1: Problem Statement and its Approximation

Andriy Bomba, Myhailo Boichura


The approach to the solving of gradient problems of parameters identification of quasiideal fields with using applied quasipotential tomographic data based on numerical complex analysis methods is transferred to cases of anisotropic media. We, similar to the existing works of world scientists, some additional information about the nature of the distribution of conductivity inside the domain (research object) is considered a priori known. However, in opposite to the traditional approaches to the statement and solving the problems of electrical impedance tomography, we set the local velocities distribution of a substance (liquid, current) in addition to the averaged potential at the contact sections of plate and body and at other sections (stream lines), we set the potential distribution (according to experimental data, which we approximate using splines, Bezier curves, etc.). Generation of initial data at the boundary of the investigated object is carried out in accordance with the polar model of current injection and a given sum of eigenvalues of the conductivity tensor of the medium. The presence of this kind of data greatly accelerates the process of further solving the problem, which is convenient, in particular, when verifying the method that developed by authors. The corresponding problem is reduced to the iterative solving of a series of problems for the Laplace type equations, where instead of «boundary numerical analogues of the Cauchy-Riemann type equations» appear the ratio of quasiorthogonality with using special types of optimization conditions. In particular: the minimizing functional is constructed by taking into account the Cauchy-Riemann type conditions, the relation between eigenvalues of corresponding anisotropy tensor and also regularizing term; the condition-restriction is built based on ellipticity conditions.

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