Mathematical Sciences: Projects in Inverse Conductivity

八名本科生将获得本科生研究经验奖。他们将在两位指导老师的指导下研究与逆电导率有关的数学问题。这类问题的原型是只知道一个物体的表面电压和电流就能求出它的电导率。这个问题的离散（和非常困难的）模拟发生在用电阻网络代替导电介质时。如果边界电流已知，则存在一种确定电阻的算法。学生们要解决的问题之一是从电势中确定边界电流。有了这些，他们将继续考虑诸如确定算法工作的最大网络的大小，检测零或无穷大电导率的电阻等问题。此外，将努力从表面电流中确定区域的形状，以估计用离散网络代替连续介质所获得的近似程度。许多其他相关的问题自然会出现，这将为学生提供现实的挑战，同时协助高级研究员进行他们正在进行的研究项目。

Eight undergraduate students will be supported by this Research Experiences for Undergraduates award. They will work under the direction of two faculty advisors on mathematical problems concerned with inverse conductivity. The prototype of such problems is that of finding electrical conductivity of a body knowing only the surface voltages and currents. The discrete (and very difficult) analogue of this problem occurs when one replaces the conducting medium with a network of resistors. If the boundary current is known, then an algorithm exists which determines the resistances. One of the problems the students will work on is that of determining the boundary current from the potential. With that, they will go on to consider questions such as determining the size of the largest network for which the algorithm works, the detection of resistors of zero or infinite conductivity. In addition, work will be done in an effort to determine the shape of the region from surface currents, in estimating the degree of approximation one gets by replacing continuous media by a discrete network. Many other related problems will arise naturally which will provide realistic challenges to the students while assisting the senior investigators with their on-going research programs.