Local Dirichlet - Neumann map and the reconstruction algorithm

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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