Publication Date

8-2016

Date of Final Oral Examination (Defense)

5-18-2016

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Supervisory Committee Chair

Grady Wright, Ph.D.

Supervisory Committee Co-Chair

Alex Townsend, Ph.D.

Supervisory Committee Member

Jodi L. Mead, Ph.D.

Abstract

A new low rank approximation method for computing with functions in polar and spherical geometries is developed. By synthesizing a classic procedure known as the double Fourier sphere (DFS) method with a structure-preserving variant of Gaussian elimination, approximants to functions on the sphere and disk can be constructed that (1) preserve the bi-periodicity of the sphere, (2) are smooth over the poles of the sphere (and origin of the disk), (3) allow for the use of FFT-based algorithms, and (4) are near-optimal in their underlying discretizations. This method is used to develop a suite of fast, scalable algorithms that exploit the low rank form of approximants to reduce many operations to essentially 1D procedures. This includes algorithms for differentiation, integration, and vector calculus. Combining these ideas with Fourier and ultraspherical spectral methods results in an optimal complexity solver for Poisson's equation, which can be used to solve problems with 108 degrees of freedom in just under a minute on a laptop computer. All of these algorithms have been implemented and are publicly available in the open-source computing system called Chebfun [21].

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