Local Dirichlet - Neumann map and the reconstruction algorithm

研究了由物体表面电压和电流的测量值反演物体电导率的反问题。在数学上，这被表述为从边界上的解的知识确定一些椭圆方程的系数的问题。这在医学科学中具有重要的应用，以通过放置在患者身体上的电极测量弱电流来确定肿瘤的位置，并且在非破坏性技术问题中也具有重要的应用。首次建立了当电导率与背景材料相比较大时确定电导率不连续部分位置的理论，并给出了数值计算的算法。在芬兰Samuli Siltanen博士和日本两位数值分析家Ide博士和Nakata博士的合作下，我们利用解析公式对二维矩形区域和半圆形区域进行了数值计算， ...更多信息 用有限元法。我们还建立了与Barber-Brown算法相关的电导率重建的数学理论。这是非常重要的，因为这个算法是已知的，虽然它的数学背景是未知的有效。三维双曲空间中时球上的边值问题在研究该算法中起着重要作用。我们的部分结果在日本数学学会年会上，在理论和应用力学会议上，也在反问题会议在英国举行。Kakehi与Gonzalez一起研究了仿射Grassmanian流形上的Radon变换。Kametaka研究了Sobolev不等式中的最佳常数。为了代表这些结果，并交流有关最近发展的信息，我们组织了一个“数学分析研讨会”的反问题，有20名来自日本的参与者和5名外国研究人员。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