Inverse Boundary Value and Inverse Scattering Problems

当人们希望在不穿透物体内部的情况下，从远程观测获得的数据或从表面测量获得的数据中恢复物体的物理性质时，就会出现逆散射问题。从广义上解释，它们包括无损检测和医学成像方法。这里提出的问题包括表明，原则上有可能从非均匀弹性体对其表面压力的响应来确定其性质，以及从反射声波来确定围绕固体障碍物和障碍物本身的介质的声学性质。更具体地说，我们提出的问题包括：1)表明可以唯一地确定非均匀弹性体对无穷小边界变形的响应的Lame参数。2)对于非均匀介质中的障碍物，我们提出可以通过固定频率的散射振幅来恢复介质中的障碍物和声速。3)在技术上更困难的二维空间情况下做问题2)，在那里它成为一个“确定的”逆问题。

PI: James Ralston/Gregory EskinDMS-0139192Abstract Inverse scattering problems arise when one wishes to recover physical properties of anobject from data obtained by remote observations or from measurements made on its surface without penetrating the interior. Interpreted broadly they include methods of nondestructive testing and medical imaging. The problems proposed here include showing that it is possible, in principle, to determine the properties of an inhomogeneous elastic body from its response to pressures on its surface, and to determine both the acoustic properties of a medium surrounding a solid obstacle and the obstacle itself from reflected sound waves.More specifically, the problems we propose include:1) showing that it is possible to uniquely determinethe Lame' parameters of an inhomogeneous elastic bodyfrom its response to infinitesimal boundary deformations.2) For an obstacle in an inhomogeneous medium we propose to show that both the obstacle and the sound speed in the medium can be recovered from the scattering amplitude ata fixed frequency.3) Do problem 2) in the more technically difficult caseof two space dimensions, where it becomes a "determined"inverse problem.