Inverse Boundary Problems for an Anisotropic Riemannian Polyhedron

各向异性黎曼多面体的逆边界问题

基本信息

批准号：
EP/D065771/1
负责人：
Anna Kirpichnikova
金额：
$ 28.16万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2006
资助国家：
英国
起止时间：
2006 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FD065771%2F1
关键词：
Inverse Boundary Problems Anisotropic Riemannian

项目摘要

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).

在直接测量是不可能或非常昂贵的情况下，存在许多问题，需要得出有关某些对象属性的结论。解决这些问题的唯一方法是使用与外部对象相关的物理领域的测量。这种问题称为反问题。有很多培养基由各种组成部分组成，即多培养基介质，例如，人体由肌肉，骨骼，脂肪和其他组织组成。同样在地球物理中，地球由岩石，粘土等组成。这些组件中的每一个的特征都以与各种米数相同的参数进行特征，因此这些参数在公共零件 - 接口上具有跳跃（不连续性）。除此之外，在现实世界中，这些组件通常是各向异性的（介质特性不仅取决于点，而且取决于方向），例如岩石，晶体，肌肉等。尽管这些问题非常重要，但现在几乎无法解决。有几种方法可以解决一种组件媒体解决各向同性逆问题或反问题，而各向异性多组分物体几乎没有。我项目的主要方向是从不同类型的边界数据中找到各向异性多组分介质未知属性的过程。我还计划证明这些反问题的唯一定理。我项目的第二部分是开发从两种类型的边界数据（测量）重建一般各向同性多组分机体的数值算法。