Inverse Scattering for Obstacles and Related Problems

当给定的数据是固定能量下的散射幅度时，提出者将处理反系数问题。期望的结果是，这确定了未知系数。具体地说，他们将考虑外部区域中的声波方程的这个问题，该区域的声速和边界是由散射数据确定的未知数。他们还对从Laplace-Beltrami算子的散射数据中恢复度量模-微分同态的相关问题感兴趣。所提出的攻击方法是通过他们在早期工作中从Faddeev型格林函数导出的积分方程来实现的。前面的问题将从三个或更多个维度提出。在两个维度上，这种方法需要与d-bar方程等其他方法相结合。提出者将对各向异性介质中的波动方程进行研究。当人们希望从远程观测获得的数据中恢复物体的物理性质时，逆散射问题就产生了。广义地说，它们包括石油勘探和医学成像方法。本项目要研究的问题包括：理论上可以从反射声波中确定固体障碍物周围介质的声学性质和障碍物本身的形状。

The proposers will work on inverse coefficient problems where the given data is the scattering amplitude at fixed energy. The desired result is that this determines the unknown coefficients. Specifically,they will consider this problem for the acoustic wave equation inan exterior domain with the sound speed and the boundary of the domain as unknowns to be determined by the scattering data. They arealso interested in the related problem of recovering metrics modulodiffeomorphisms from scattering data for Laplace-Beltrami operator.The proposed method of attack is via integral equations that they derived from Faddeev-type Green's functions in earlier work. The preceding problems will be posed in three or more dimensions. In twodimensions this approach needs to be combined with other methods likethe d-bar equation. The proposers will investigate ways of doing thisfor the wave equation in an anisotropic medium. Inverse scattering problems arise when one wishes to recover physicalproperties of an object from data obtained by remote observations.Interpreted broadly, they include methods oil exploration and medicalimaging. The problems that will be studied in this project include showing that it is theoretically possible to determine both the acousticproperties of a medium surrounding a solid obstacle and the shape of the obstacle itself from reflected sound waves.