Publication Date

8-2017

Date of Final Oral Examination (Defense)

6-2-2017

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Supervisory Committee Chair

Grady B. Wright, Ph.D.

Supervisory Committee Member

Jodi Mead, Ph.D.

Supervisory Committee Member

Donna Calhoun, Ph.D.

Abstract

Radial basis functions (RBFs) were originally developed in the 1970s for interpolating scattered topographic data. Since then they have become increasingly popular for other applications involving the approximation of scattered, scalar-valued data in two and higher dimensions, especially data collected on the surface of a sphere. In the late 2000s, matrix-valued RBFs were introduced for approximating divergence-free and curl-free vector fields on the surface of a sphere from scattered samples, which arise naturally in atmospheric and oceanic sciences. The intriguing property of these RBFs is that the resulting vector-valued approximations analytically preserve the divergence-free or curl-free properties of the field.

The most commonly used RBFs feature a shape parameter that controls how peaked or flat the basis functions are, with the choice of this parameter greatly affecting the accuracy of the RBF approximation to the underlying data. Flatter basis functions, which correspond to small shape parameters, generally result in more accurate approximations when the sampled data comes from a smooth function or vector-field. However, the direct method for computing the resulting RBF approximation becomes horribly ill-conditioned as the basis functions are made flatter and flatter. For scalar-valued RBF approximation, this was a fundamental issue until the mid-2000s when researchers started to develop stable algorithms for "flat" RBFs. One of the most successful of these is the RBF-QR algorithm, which completely bypasses the ill-conditioning associated with flat scalar-valued RBFs on the sphere using a clever change of basis. In this thesis, we extend the RBF-QR algorithm to flat matrix-valued RBFs for approximating both divergence-free and curl-free vector fields on the sphere. We give numerical results illustrating the effectiveness of this new algorithm and also show that in the limit where the matrix-valued RBFs become entirely flat, the resulting approximations converge to vector spherical harmonic approximants. This is the first algorithm that allows for stable computations of divergence-free and curl-free matrix-valued RBFs in the flat limit.

DOI

https://doi.org/10.18122/B2G70C

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