Far Fields and Remote Sensing

University Of Washington, Seattle WA

Investigators

Abstract

DMS-0355455 Title: Far fields and Remote sensing PI: John Sylvester (University of Washington) ABSTRACT We study inverse scattering and inverse source problems for the Helmholtz equation, Maxwell's equations, and other PDE's. Our efforts focus on identifying those characteristics of size and location that can be computed from data sets that are much too small to uniquely determine the source or scatterer. The Born approximation to time harmonic scattering is the Fourier Transform. In this approximation, one such data set would be the restriction of the Fourier transform of a function to a sphere . In this case, it is impossible to find the support of the function from this data, but we can find a smallest convex set, a set which must belong to the convex hull of the support of any function with this restricted Fourier transform. This set can be stably calculated, even from noisy data. For other data sets, there are other "minimal supports", where the definition of "support" depends on the data set. We seek to discover the correct notion of support for different data sets and different models. In the presence of a priori information (e.g. polygonal shape or positivity), these minimal sets may even describe true supports. The focus of this project is to develop and improve algorithms for acoustical and electro-magnetic remote sensing (e.g. radar, sonar, ultrasound). The mathematical theory of inverse scattering includes powerful methods for deducing the material properties of an object from data measured in a remote sensing experiment. However, there are many practical applications where it is impossible to make observations at sufficiently many frequencies and angles to apply these methods. In this project, we focus on understanding those aspects of size and location that can be unequivocally identified from limited data sets, and developing algorithms to stably compute these sizes and locations.