Date of Award

Fall 2025

Project Type

Dissertation

Program or Major

Applied Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Mark E Lyon

Second Advisor

John McHugh

Third Advisor

Kai Germaschewski

Abstract

Boundary integral equation (BIE) methods are advantageous in scattering problems due to their reduction of the problem's dimensionality as well as their automatic satisfaction of far-field radiation conditions. Specifically, for electromagnetic scattering off of perfectly electrically conducting (PEC) geometries, we are concerned with the magnetic field integral equation (MFIE). Typically, BIE implementation is iterative, with each direct integration requiring $\mathcal{O}(N^2)$ operations for a grid size $N$. For large $N$, this incurs very large computation times. To date, all existing methods that reduce the asymptotic operational complexity of evaluating the MFIE integral to $\mathcal{O}(N\log N)$ rely on use of the Fast Fourier Transform (FFT). For oscillatory integral kernels, hardware limitations cap the speed at which FFT information is exchanged in bottle-necking the overall speed-up. To remedy this, this thesis successfully adapts Bruno and Bauinger's interpolated factored Green's function (IFGF) algorithm to the MFIE, which achieves $\mathcal{O}(N\log N)$ operational complexity by way of interpolating a slow-varying factor of the integral kernel without the use of FFTs. We present a full description of the algorithm, followed by numerical results that demonstrate the accuracy and speed-up of the IFGF implementation. For example, a scattering problem of over 1.5 million unknowns achieves a 52-times speed-up while maintaining an accuracy of the order of $10^{-5}$. Several accelerated numerical experiments on physically interesting geometries are completed as well.

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