Inverse Boundary Problems for an Anisotropic Riemannian Polyhedron

在直接测量不可能或非常昂贵的情况下，存在大量需要对某些对象的性质进行推断的问题。解决这些问题的唯一方法是使用与外部物体相关的物理场的测量。这类问题称为反问题。有很多介质是由各种成分组成的，即多组分介质，例如，人体由肌肉、骨骼、脂肪等组织组成；在地球物理中，地球也由岩石、粘土等组成。对于不同的材料，这些部件的每个部件都具有本质上不同的参数，因此这些参数在公共部件-界面上具有跳跃(不连续)。此外，在现实世界中，这些分量通常是各向异性的(介质性质不仅取决于点，而且还取决于方向)，例如岩石、晶体、肌肉等。这些问题虽然很重要，但现在几乎没有解决。对于各向同性或单组分介质的反问题有多种方法，而对于各向异性的多组分体几乎没有一种方法。本课题的主要研究方向是寻找一种从不同类型的边界数据中重建各向异性多分量介质未知性质的方法。此外，我还计划证明这些反问题的唯一性定理。本项目的第二部分是开发由两种边界数据(测量值)重建一般各向同性多分量物体的数值算法。

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).