Applied Quasipotential Method for Solving the Coefficient Problems of Parameter Identification of Anisotropic Media

Abstract

A numerical method of quasiconformal mappings for solving the coefficient problems of finding eigenvalues of the conductivity tensor having information about its directions in an anisotropic medium using applied quasipotential tomographic data is generalized. The corresponding algorithm is based on the alternate solving of problems on quasiconformal mappings and parameter identification. The results of numerical experiments of imitative restoration of environment structure are presented.

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

“…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 Introduction. As it is known [1][2][3][4][5], the image reconstruction of an isotropic conductivity coefficient that based on the applied quasipotential tomography (AQT) requires the imposition of a large number of conditions at the domain bound, and also the structure of the corresponding medium. It turns out that in the general case (in contrast to some specific [6]) of anisotropy, it is necessary to set much more information about the conductivity distribution for its parameter identification [1,[7][8][9][10][11][12].…”

“…This, obviously, weakens the correctness of the problem in comparison with the isotropy. And, consequently, it requires (in comparison with, for example, [5]) the necessity of using a regularizing functional, in particular the Tikhonov type [1,3,4,9,11]. The ways to apply additional data about the conductivity tensor (CT), depending on the information type about it are proposed in a number of papers [6][7][8][9]12].…”

“…On the other hand, today a promising methodology for identifying the conductivity coefficient using AQT data, according to which the solving of the sequence of so-called analysis and synthesis problems is reduced to the alternate application of numerical quasiconformal mappings methods and the parameters identification of the conductivity of the medium, respectively [4,5] is developed. In this paper we discuss the transfer of this methodology to the case of the parameter identification of anisotropic media.…”

“…% % is given an initial starting point}, generated by the existence of [4,5], by using sets of values ( ) ( ) (…”

“…We denote the corresponding for current injection bound of domain z G with given four marked points by As a mathematical model of AQT [1] we use, similarly to [4,5,14], systems of differential equations in partial derivatives which connect mutually quasicomplex conjugate quasipotentials…”