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Mathematical analysis of scattering phenomena and inverse problems

散射现象及反问题的数学分析

基本信息

批准号：
13440048
负责人：
ISOZAKI Hiroshi
金额：
$ 5.31万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

This reserach project aimed at the development of the study of inverse problems arising from the mathematical analysis of scattering phenoma. The main theme was the study of spectra of differential operators. As a main conference project, we have organized an international workshop of inverse problems on October 2002 at Kyoto, where leading reserchers of this field came together, and promoted much interest on this filed in Japan.The head investigator proposed a new method for solving multi-dimensional inverse problems. The essential idea consists in embedding the inverse boundary value problem in R^n to the hyperbolic space. This new method introduced new view points of inverse problems. He discussed the inverse problem for the local perturbation of conformal metrics on hyperbolic manifolds. As an interestring by-product, he proved that in the boundary value problems in R^3, the electric conductivities can be identified locally from the knowledge of local Dirichlet-Neumann map. This hyperbolc space approach can also be applied to the problem of identification of locations of inclusions, and the reconstruction problem for linearized equations. They are expected to have applications to medical science. Okada studied numerical harmonic analysis, in particular, spline functions, wavelet analysis and numerical computation. Mochizuki studied inverse problems of reconstructing coefficients of Dirac operators from the data in finite intervals. Yoshitomi studied the properties of eigenvalues of the Laplacian on the 2-dimensional domain with crack, and also those of band region. Nakamaura studied the inverse problem of identifying the obstacle from the refelcted waves and also the inverse problem for elastic equations. Tamura studied the Aharonov-Bohm effect for the 2-diemnsional Schrodinger operators with Dirac type magnetic fields.

本研究旨在发展由散射现象的数学分析所引起的反问题的研究。主要的主题是研究谱的微分算子。2002年10月在日本的京都举办了一次国际反问题研讨会，作为会议的主要项目之一，该研讨会汇集了该领域的主要研究人员，促进了日本对该领域的兴趣，主要研究人员提出了一种求解多维反问题的新方法。其基本思想是将R ^n中的反边值问题嵌入到双曲空间中。这种新方法引入了反问题的新观点。他讨论了反问题的局部扰动共形度量双曲流形。作为一个有趣的副产品，他证明了在R^3中的边值问题中，电导率可以从局部狄利克雷-诺依曼映射的知识中局部地确定出来。这种双曲空间方法也可以应用于夹杂物位置的识别问题，以及线性化方程的重构问题。它们有望在医学上得到应用。冈田研究数值谐波分析，特别是样条函数，小波分析和数值计算。Mochizuki研究了由有限区间数据重构Dirac算子系数的反问题。Yoshitomi研究了二维裂纹区域上的Laplacian特征值的性质，也研究了带状区域的特征值的性质。Nakamaura研究了从反射波中识别障碍物的反问题以及弹性方程的反问题。Tamura研究了Dirac型磁场作用下二维Schrodinger算子的Aharonov-Bohm效应。