Date of Award

Spring 1-1-2012

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Gregory Beylkin

Second Advisor

Gunnar Martinsson

Third Advisor

Keith Julien

Fourth Advisor

Francois Meyer

Fifth Advisor

Rafael Peistun

Abstract

This thesis consists of contributions to three topics: algorithms for computing generalized Gaussian quadratures, tomographic imaging algorithms, and reduction algorithms. Our approach is based on using non-linear approximations of functions. We develop a new algorithm for constructing generalized Gaussian quadratures for exponentials inte- grated against a non-sign-definite weight function. These quadratures integrate band-limited exponentials to a user-defined accuracy. We also introduce a method of computing quadrature weights via l∞ minimization. Second, we develop a new imaging algorithm for X-ray tomography. This algorithm, Polar Quadrature Inversion, uses rational approximations to approximate tomographic projections with a near optimal number of terms for a given accuracy. This rational signal model allows us to augment the measured data by extending the tomographic projection's domain in Fourier space. As the extended data from all the projections fill a disk in the Fourier domain, we use polar quadratures for band-limited exponentials and the Unequally Spaced Fast Fourier Transform to obtain our image. We demonstrate that the resulting images have significantly improved resolution without additional artifacts near sharp transitions. Finally, we develop an extension of existing reduction algorithms for functions of one variable to functions of many variables. By reduction, we understand an approximation (to a user-supplied accuracy) of a linear combination of decaying exponentials by a representation of the same form but with a minimal number of terms. While for functions of one variable there is an underlying theory based on the analysis of functions of one complex variable, no such theory is available for the multivariate case. Our approach is a first step in the development of such theory. We demonstrate our algorithm on two examples of multivariate functions, a suboptimal linear combination of real-valued, decaying exponentials, and that of complex-valued, decaying exponentials.

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