Inverse Boundary Problems for an Anisotropic Riemannian Polyhedron

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There exist a great number of problems that require to conclude about the properties of some objects in the case when the direct measurements are impossible or very expensive. There is the only way to tackle those problems, this way is to use the measurements of physical fields related to those objects outside of them. Problems of that kind are known as inverse problems. There are a lot of media that consist of various components, i.e. multi-componet media, for instance, human bodies consist of musclus, bones, fat and other tissues; also in geophysics, earth consists of rocks, clay,...etc. Each of these components is characterized by parameters that differs essentialy for various meterials, thus these parameters have jumps (discontinuities) on the common parts - interfaces. Besides that in real world these components are usually anisotropic (the medium properties depend not only on the point but also on the direction), for example, rocks, crystal, muscules,etc. Although these problems are of a great importance they are almost unsolved now. There are several methods to tackle isotropic inverse problems or inverse problems for one component media, and almost none for anisotropic multi-component bodies. The main direction of my project is to find a procedure of the reconstruction of unknown properties of the anisotropic multi-component medium from different types of boundary data. Also I plan to prove the uniqueness theorems for these inverse problems. The second part of my project is to develop numerical algorithm for the reconstruction of the general isotropic multi-component body from two types of boundary data (measurements).