Inverse Boundary Value and Inverse Scattering Problems

Pi：James Ralston/Gregory EskinDMS-0139192当人们希望从通过远程观测获得的数据或从不穿透内部的物体表面进行的测量中恢复物体的物理性质时，出现了抽象的逆散射问题。广义的解释包括无损检测和医学成像方法。这里提出的问题包括，原则上可以根据非均匀弹性体对其表面压力的响应来确定其性质，并根据反射声波来确定固体障碍物周围介质的声学性质和障碍物本身的声学性质。更具体地说，我们提出的问题包括：1)证明可以从非均匀弹性体对无限小边界变形的响应中唯一地确定其Lame参数。2)对于非均匀介质中的障碍物，我们建议证明障碍和介质中的声速都可以从固定频率的散射幅度中恢复。3)在技术上更困难的两个空间维的情况下，做问题2)在那里它变成了一个“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.