Far Fields and Remote Sensing

点击翻译按钮获取中文摘要

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.