Local Dirichlet - Neumann map and the reconstruction algorithm

研究了通过测量物体表面的电压和电流来重建物体电导率的反问题。在数学上，这被表述为根据边界上解的知识确定某些椭圆方程系数的问题。这在医学科学中有重要的应用，通过放置在病人身上的电极测量微弱电流来确定肿瘤的位置，也在非破坏性的技术问题中。我们首先建立了确定电导率相对于背景材料较大时不连续部分位置的理论，并找到了数值计算的算法。在芬兰samuli Siltanen博士和日本数值分析专家Dr.Ide和Dr.Nakata博士的合作下，我们分别在二维矩形域和半圆域用解析公式进行了数值计算，然后用有限元法进行了数值计算。结果非常好，证明了我们的想法是有效的。我们还构建了与著名的Barber-Brown算法相关的数学理论，用于电导率的重建。这是非常重要的，因为这个算法是已知有效的，尽管它的数学背景是未知的。在研究该算法的过程中，主要考虑了三维双曲空间中星形球的边值问题。我们的部分成果在日本数学学会年会上、理论与应用力学会议上以及在英国举行的逆问题会议上发表。Kakehi和Gonzalez一起研究了仿射格拉斯曼流形上的Radon变换。Kametaka研究了Sobolev不等式中的最佳常数。为了展示这些成果，并就最近的发展交流信息，我们组织了一个关于逆问题的“数学分析研讨会”，有20名来自日本的参与者和5名外国研究人员参加。少

We studied the inverse problem of reconstructing the electric conductivity of a body from the measurement of the voltage and current on the surface. Mathematically, this is formulated as the problem of determining the coefficients of some elliptic equation from the knowledge of the solution on the boundary. This has important applications in medical science to determine the location of tumor by the measurement of the weak current by the electrodes put on the body of the patient, and also in non-destructive technological problems. We first established the theory to determine the location of the discontinuous part of the electric conductivity when it is large compared to the back ground material, and found the algorithm of numerical computation. Under the collaboration of Dr.Samuli Siltanen from Finland, and two Japanese numerical analysts, Dr.Ide and Dr.Nakata, we did numerical computation in 2-dimensional rectangular domain, and semi circular domain by using analytical formula and then … More by the finite element method. The result is extremely good and proves the efficiency of our idea.We also constructed the mathematical theory related with the well-known Barber-Brown algorithm for the reconstruction of the electric conductivity. This is very significant, since this algorithm is known to be effective although its mathematical background was unknown. To study this algorithm the important role is played by the boundary value problem in the horosphere in 3-dimensional hyperbolic space. Some parts of our results were presented in the annual meeting of the Japanese Mathematical Society, in the conference of theory and applied mechanics, and also in the conference on inverse problems held in England. Kakehi studied the Radon transform on Affine Grassmanian manifolds with Gonzalez. Kametaka studied the best constant in the Sobolev inequality. To represent these results and also to exchange information on the recent developments, we organized a "Mathematical Analysis Seminar" on the inverse problem with 20 participants from Japan and also 5 foreign researchers. Less