Far Fields and Remote Sensing

题目：远场与遥感pi: John Sylvester （University of Washington）摘要我们研究了helmholtz方程、Maxwell方程和其他偏微分方程的逆散射和逆源问题。我们的工作重点是识别那些大小和位置的特征，这些特征可以从数据集计算出来，这些数据集太小，无法唯一地确定源或散射。时间谐波散射的玻恩近似是傅里叶变换。在这个近似中，一个这样的数据集将是一个函数的傅里叶变换对球面的限制。在这种情况下，我们不可能从这些数据中找到这个函数的支持，但是我们可以找到一个最小的凸集，这个集合必须属于这个受限傅里叶变换的任何函数的支持的凸包。即使从有噪声的数据中也可以稳定地计算出该集合。对于其他数据集，还有其他的“最小支持”，其中“支持”的定义取决于数据集。我们试图发现支持不同数据集和不同模型的正确概念。在存在先验信息的情况下（例如多边形形状或正性），这些最小集甚至可以描述真支持。该项目的重点是开发和改进声学和电磁遥感（如雷达、声纳、超声波）的算法。逆散射的数学理论包括从遥感实验中测量的数据推断物体的材料特性的有效方法。然而，在许多实际应用中，不可能在足够多的频率和角度进行观测来应用这些方法。在这个项目中，我们专注于理解那些可以从有限的数据集中明确识别的大小和位置的方面，并开发算法来稳定地计算这些大小和位置。

DMS-0355455Title: Far fields and Remote sensingPI: John Sylvester (University of Washington)ABSTRACTWe study inverse scattering and inverse source problems for theHelmholtz equation, Maxwell's equations, and other PDE's. Our effortsfocus on identifying those characteristics of size and location thatcan be computed from data sets that are much too small to uniquelydetermine the source or scatterer.The Born approximation to time harmonic scattering is the FourierTransform. In this approximation, one such data set would be therestriction of the Fourier transform of a function to a sphere . Inthis case, it is impossible to find the support of the function fromthis data, but we can find a smallest convex set, a set which mustbelong to the convex hull of the support of any function with thisrestricted Fourier transform. This set can be stably calculated, evenfrom noisy data.For other data sets, there are other "minimal supports", where thedefinition of "support" depends on the data set. We seek to discoverthe correct notion of support for different data sets and differentmodels. In the presence of a priori information (e.g. polygonal shapeor positivity), these minimal sets may even describe true supports. The focus of this project is to develop and improve algorithms foracoustical and electro-magnetic remote sensing (e.g. radar, sonar,ultrasound).The mathematical theory of inverse scattering includes powerfulmethods for deducing the material properties of an object from datameasured in a remote sensing experiment. However, there are manypractical applications where it is impossible to make observations atsufficiently many frequencies and angles to apply these methods. Inthis project, we focus on understanding those aspects of size andlocation that can be unequivocally identified from limited data sets,and developing algorithms to stably compute these sizes and locations.