Analysis of Anisotropic Inverse Boundary Value Problems

各向异性效应在无损检测、电子物理学和医学成像中的电学、声学和光学成像中是重要的。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，标量各向异性反问题的理论成果很少，没有一般的方法对稳定性和发展收敛的重建算法来解决这些问题。几乎没有什么是已知的偏微分方程组的反边值问题，特别是在各向异性的情况下。本研究的目的是开发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，并通过开发收敛的重建算法来解决完全非线性反边值问题来实现这些方法。

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.