Analysis of Anisotropic Inverse Boundary Value Problems

各向异性的影响在无损检测、地球物理和医学成像中使用的电、声和光学成像中很重要。如果忽略这些影响，可能会导致反问题的错误解决方案。目前关于标量各向异性逆问题的理论成果很少，也没有一般性的方法来解决这些问题的稳定性和收敛重建算法的发展。对于偏微分方程组的边值反问题，特别是各向异性的边值反问题，人们几乎一无所知。本研究的目的是发展分析和几何方法来研究各向异性反问题的唯一性和稳定性，并通过开发求解全非线性反边值问题的收敛重建算法来实现这些方法。

Effects of anisotropy are important in electrical, acoustical and optical imaging used in non-destructive testing, geophysics and medical imaging. If ignored these effects can result in a wrong solution to the inverse problem. There are currently very few theoretical results on scalar anisotropic inverse problems with no general approach towards stability and development of convergent reconstruction algorithms for solving these problems. Almost nothing is known about inverse boundary value problems for systems of partial differential equations, especially in the anisotropic case. The aim of this research is to develop analytical and geometric methods to study uniqueness and stability in anisotropic inverse problems and to implement these methods by developing convergent reconstruction algorithms for solving fully non-linear inverse boundary value problems.