Kernel-Based Galerkin Methods on Compact Manifolds Without Boundary, with an Emphasis on SO(3)

dc.contributor.advisor Hangelbroek, Thomas
dc.contributor.author Collins, Patrick
dc.contributor.department Mathematics
dc.date.accessioned 2021-09-30T18:15:15Z
dc.date.available 2021-09-30T18:15:15Z
dc.date.issued 2021
dc.description.degree Ph.D.
dc.identifier.uri http://hdl.handle.net/10125/76422
dc.subject Mathematics
dc.subject Approximation Theory
dc.subject Galerkin Methods
dc.subject Kernel-Based
dc.subject Mathematics
dc.title Kernel-Based Galerkin Methods on Compact Manifolds Without Boundary, with an Emphasis on SO(3)
dc.type Thesis
dcterms.abstract We develop a kernel-based Galerkin method for numerically solving elliptic partial differential equations on compact Riemannian manifolds without boundary that are \textit{mesh-free}, \textit{coordinate-free}, \textit{regularity-preserving}, and \textit{high-performing}. A fundamental challenge is to combine a theoretical solution with mesh-free \textit{quadrature} in such a way that approximation power is not lost.\\\indent We show that the approximation power of the computed (discretized) solution can be made to be on par with the approximation power of the theoretical solution, provided the \textit{oversampling exponent} is sufficiently large. We then show how to truncate a kernel on $ SO(3) $ given by a Hilbert-Schmidt series in such a way that the approximation power of both the truncated solution and the discretized truncated solution (again using quadrature) is on par with the approximation power of the theoretical solution, provided the \textit{truncation parameter} is sufficiently large.
dcterms.extent 131 pages
dcterms.language en
dcterms.publisher University of Hawai'i at Manoa
dcterms.rights All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner.
dcterms.type Text
local.identifier.alturi http://dissertations.umi.com/hawii:11154
Files
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
Name:
Collins_hawii_0085A_11154.pdf
Size:
581.51 KB
Format:
Adobe Portable Document Format
Description: