Mathematica Sciences: Inverse Boundary Value Problems

该项目将支持偏微分方程逆问题的研究。特别是，研究将集中在阻抗层析成像问题、Dirichlet到Neumann映射的表征问题、维度大于或等于3的各向异性电导率的逆电导问题、以及逆散射中的相关问题和微分几何中出现的逆问题。当将电势施加到导电物体的表面时，产生的流过表面的电流取决于物体的内阻。因此，在物体边界上进行的电压和电流测量可以用来探测物体的内部结构。问题是通过测量边界上的电压电位和相应的电流通量来确定尽可能多的关于内电阻率的信息。这个问题，通常被称为阻抗计算机断层扫描或电阻抗成像，在地球物理学中出现，因为试图确定地球的组成。最近，它被提议作为一种潜在的有价值的医学和生物科学诊断工具。

This project will support research on inverse problems in partial differential equations. In particular the research will focus on the impedance tomography problem, the problem of characterization of the Dirichlet to Neumann map, the inverse conductivity problem for anisotropic conductivities in dimensions greater than or equal to three, and related problems in inverse scattering and inverse problems that arise in differential geometry. When an electric potential is applied to the surface of a conducting body, the resulting current flux across the surface depends on the internal resistance of the body. For this reason, voltage and current measurements made on the boundary of an object can be used to probe the internal structure of the body. The problem is to determine as much information as possible about the internal resistivity from measurements of voltage potentials and corresponding current fluxes at the boundary. This problem, often referred to as impedance computed tomography or electrical impedance imaging, arose in geophysics from attempts to determine the composition of the earth. More recently, it has been proposed as a potentially valuable diagnostic tool for the medical and biological sciences.