Inverse Scattering for Obstacles and Related Problems

提出者将研究反系数问题，其中给定的数据是固定能量下的散射振幅。期望的结果是，这决定了未知系数。具体来说，他们将考虑在声速和边界未知的情况下，由散射数据确定的外域声波方程的问题。他们还对拉普拉斯-贝尔特拉米算子散射数据中度量模微分同态恢复的相关问题感兴趣。提出的攻击方法是通过他们在早期工作中从faddeev型格林函数中导出的积分方程。上述问题将在三个或多个维度上提出。在二维中，这种方法需要与其他方法如d-bar方程相结合。提议者将研究在各向异性介质中的波动方程中这样做的方法。当人们希望从远程观测获得的数据中恢复物体的物理特性时，就会出现逆散射问题。从广义上讲，它们包括石油勘探和医学成像方法。该项目将研究的问题包括，从理论上可以通过反射声波来确定围绕固体障碍物的介质的声学特性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.