Date of Award

Spring 1-1-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Gregory Beylkin

Second Advisor

Bradley Alpert

Third Advisor

Mark Ablowitz

Fourth Advisor

Per-Gunnar Martinsson

Fifth Advisor

Rafael Piestun

Abstract

This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become computationally infeasible.

The first part describes a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter, without requiring the filter to be causal, and possibly with a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter. Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade. In this FIR approximation, the non-causal part of the IIR filter only introduces an additional delay. Because our IIR construction does not have to enforce causality, the filters we design are more efficient than filters designed by existing methods.

The second part describes a fast algorithm to propagate, for any desired accuracy, a time-harmonic electromagnetic field between two planes separated by free space. The analytic formulation of this problem (circa 1897) requires the evaluation of the Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is large, it can be accurately approximated by asymptotic methods. The computational difficulties arise in the intermediate region where, in order to obtain an accurate solution, it is necessary to apply the oscillatory Rayleigh-Sommerfeld kernel as is. In our approach, we accurately approximate the kernel by a short sum of Gaussians with complex exponents and then efficiently apply the result to input data using the unequally spaced fast Fourier transform. The resulting algorithm has the same computational complexity as methods based on the Fresnel approximation. We demonstrate that while the Fresnel approximation may provide adequate accuracy near the optical axis, the accuracy deteriorates significantly away from the optical axis. In contrast, our method maintains controlled accuracy throughout the entire computational domain.

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