Far Fields and Remote Sensing

DMS-0355455标题：远场和遥感PI：John Sylvester(华盛顿大学)摘要我们研究Helmholtz方程、Maxwell方程和其他偏微分方程组的逆散射和逆源问题。我们的努力集中在识别那些可以从太小的数据集计算出的大小和位置特征，这些数据集太小而无法唯一地确定源或散射体。时间谐波散射的Born近似是傅立叶变换。在这种近似下，一个这样的数据集将是函数到球面的傅里叶变换的限制。在这种情况下，不可能从这些数据中找到函数的支持度，但我们可以找到一个最小的凸集，这个集必须属于具有这种限制的傅里叶变换的任何函数的支持度的凸包。对于其他数据集，还有其他的“最小支持度”，其中“支持度”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(例如，多边形或正值)的情况下，这些最小集合甚至可以描述真实的支撑度。该项目的重点是发展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目标材料特性的强大方法。然而，在许多实际应用中，不可能在足够多的频率和角度上进行观测以应用这些方法。在这个项目中，我们专注于了解可以从有限的数据集中明确识别的大小和位置的那些方面，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.