APPLIED QUASIPOTENTIAL METHOD FOR SOLVING THE COEFFICIENT PROBLEMS OF PARAMETER IDENTIFICATION OF ANISOTROPIC MEDIA

. 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.


Introduction
Today, solving the problem of image reconstruction of a conductivity tensor (CT) in anisotropic media finds its application in an increasing number of areas. In particular, in robotics, geology, medicine, etc. (see, e.g., [6][7][8]). Despite the low resolution of the resulting images, geometric flexibility, harmlessness to the environment contributes to the continuation of relevant studies in order to improve the accuracy and speed of the calculations [6,8].
The aim of this work is to generalize the numerical quasiconformal mapping method [2 -4] for solving the coefficient problems of finding eigenvalues of the CT having information about its directions in an anisotropic medium using applied quasipotential tomographic (AQT) data.

The parameter identification problem of anisotropic media using AQT data
We consider the quasiideal processes of particles movement (in particular, liquids, electric charges) in a single-connected curvilinear domain (anisotropic layer or plate, which is some tomographic cross-section) z G (Fig. 1a), limited by a smooth  [1,2,4,6,[8][9][10]. Suppose that we known not only distributions of potentials, but also local velocities of matter at the same points [2,4]. In this case, the problem of parameters identification of the quasiideal stream using AQT data is traditionally reduced to finding an infinite number of functions-quasipotentials In practice, it is not possible to obtain an infinite number of data (measurements at the boundary), and therefore scientists apply a different kind of simplification (see, e.g., [1,2,4,[6][7][8][9][10] [2,4]. This, in comparison with the known world analogues (see, e.g., [1,6,8,10]), provides the possibility of both the physical providing of experiment and the application of our developed complex analysis methods. In this case, the mathematical model of AQT [8], similar to [1,2,4], we write in the form (1) and conditions: n is unit vector of outer normal; M is a running point of the corresponding curve. Functions as in [2], can be constructed by interpolating experimentally CT components with equal elements of the additional diagonal [10] are defined as follows: 2 and we consider the angles distribution function of extreme value directions of the conductivity coefficient ( , ), xy   similar to [1,8,9], a priori known. Here , , bb rk  are the parameters that are defined during the problem solving process.
The problem lies in image reconstruction of the CT. Here, the related is the calculation of the corresponding dynamic meshes and velocity fields.
We can reduce (1) - (5) to the series of more general boundary value problems on quasiconformal mapping  (Fig. 1a) onto the corresponding domains of the complex quasipotential () p G  (Fig. 1b)   dl is arc element of corresponding curve.

Synthesis of the numerical quasiconformal mapping method and ideas of alternating block parametrization
In [2,4], algorithms for numerical solving of inverse nonlinear boundary value problems on quasiconformal mappings in curvilinear quadrilateral domains bounded by stream and equipotential lines are proposed, and in [3] such approaches are generalized to the case of anisotropy. Accordingly, solving the problem will be carried out applying these methods (using the corresponding notations; the algorithms obtained in the abovementioned works will be fragments of wider structures, in particular, injectivity must be taken into account).
We reconstruct the CT, like in [2][3][4], provided minimize the residual sum of squares of expressions, obtained from Cauchy-Riemann-type conditions, with applying the ideas of regularization and having positive eigenvalues condition. ( where  is regularization parameter.
We write the corresponding difference analogues in the mesh  [3,12]. Namely: we set the number of injections A visual representation of received CT distribution is carried out similar to [10] by application of a specially developed procedure. According to it, the researched domain is divided into square sections by lines, which are parallel to the coordinate axes. The CT is characterized in the center of this figures as an ellipses (axes and radii of which correspond to the directions of eigenvectors and proportional to the values of eigenvalues, respectively) of the form  Fig. 2b presents the reconstructed image of the CT distribution in comparison with the given theoretically (Fig. 2a).

Acknowledgment
The numerical quasiconformal mapping method for solving the problem of finding the eigenvalues of the CT having information about its directions in anisotropic medium and using AQT data is generalized. The algorithm for solving the corresponding problem is based on the application of the idea of quasiconformal similarity in the small for construction of curvilinear quadrilaterals (quasiparallelograms), that are components of dynamic mesh in the physical domain and the corresponding squares in the domain of complex quasipotential and alternate parametrization of internal nodes of dynamic meshes (constructed for each of the injections) and the desired CT. The developed algorithm is characterized by comparatively fast computer convergence (since, unlike many useful methods, it does not require finding the derivatives of the distribution function of the CT at the specified points and specifying the boundary nodes at each iteration step), The latter may occur in areas with large computational errors (so-called "stagnant zones" and "large gradient zones") that arise near the singular points of the non-smooth boundary lines and the critical points of the interior of the corresponding domains. But we note that the considerably new algorithm is to take into account the conditions of "anisotropic quasiorthogonality" along the boundary equipotentials and flow lines (instead of orthogonality in cases of isotropic media), which causes additional substantially new constructions.
Also, the anisotropy tensor on orders affects the deterioration of accuracy, stability, which in particular requires the creation of new structures, procedures of regularization Tikhonov type.It is also worth noting that, unlike the traditional approaches to the formulation and solving the problems of electrically impedance tomography [1, 6 -10], we determine the distribution of local velocities of a substance (fluid, current) and averaged potentials on sites the contact of the plate and body, and in other sections the distribution of the potential (according to experimental data). This, of course, provides greater mathematical conformity, and therefore a certain gain in the iterative reconstruction process.
We plan to extend this algorithm to the following cases: spatial reconstruction possibility, formation the several sections of apply the quasipotential at the initial stream, parameters identification of the CT of piecewise-homogeneous and piecewise-inhomogeneous media and filtration-convectiondiffusion type processes (see, for example, [5,11]).