Approximation and interpolation employing radial basis functions has
found important applications since the early 1980's in areas such
as signal processing, medical imaging, as well as neural networks.
Several applications demand that certain physical properties be
fulfilled, such as a function being divergence free. No such class
of radial basis functions that reflects these physical properties
was known until 1994, when Narcowich and Ward introduced a family of
matrix-valued radial basis functions that are divergence free. They
also obtained error bounds and stability estimates for interpolation
by means of these functions. These divergence-free functions are
very smooth, and have unbounded support. In this thesis we
introduce a new class of matrix-valued radial basis functions that are
divergence free as well as compactly supported. This leads to the
possibility of applying fast solvers for inverting interpolation
matrices, as these matrices are not only symmetric and positive
definite, but also sparse because of this compact support. We develop
error bounds and stability estimates which hold for a broad class of
functions. We conclude with applications to the numerical solution of
the Navier-Stokes equation for certain incompressible fluid flows.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/21 |
| Date | 30 September 2004 |
| Creators | Lowitzsch, Svenja |
| Contributors | Narcowich, Francis J., Ward, Joseph D. |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | 1064009 bytes, 154392 bytes, electronic, application/pdf, text/plain, born digital |
| Rights | I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Texas A&M University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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