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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).

在直接测量不可能或非常昂贵的情况下，存在着大量的问题，需要对某些对象的属性进行推断。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。这类问题被称为逆问题。有很多介质是由各种成分组成的，即多成分介质，例如，人体由肌肉，骨骼，脂肪和其他组织组成;在地球物理学中，地球由岩石，粘土，…这些组件中的每一个的特征在于对于各种材料本质上不同的参数，因此这些参数在公共部分-界面上具有跳跃（不连续）。此外，在真实的世界中，这些组分通常是各向异性的（介质的性质不仅取决于点，而且取决于方向），例如，岩石，晶体，肌肉等。对于各向同性体和单组分体的反问题，已有多种求解方法，而对于各向异性多组分体的反问题，几乎没有求解方法。本课题的主要研究方向是从不同类型的边界数据出发，寻找一种各向异性多组分介质未知性质的重建方法。同时，我也计划证明这些反问题的唯一性定理。我的项目的第二部分是发展的数值算法的重建一般各向同性多组分体从两种类型的边界数据（测量）。