Description
I focus on algorithms that generate good sampling points for function approximation. In 1D, it is well known that polynomial interpolation using equispaced points is unstable. On the other hand, using Chebyshev nodes provides both stable and highly accurate points for polynomial interpolation. In higher dimensional complex regions, optimal sampling points are not known explicitly. This work presents robust algorithms that find good sampling points in complex regions for polynomial interpolation, least-squares, and radial basis function (RBF) methods. The quality of these nodes is measured using the Lebesgue constant. I will also consider optimal sampling for constrained optimization, used to solve PDEs, where boundary conditions must be imposed. Furthermore, I extend the scope of the problem to include finding near-optimal sampling points for high-order finite difference methods. These high-order finite difference methods can be implemented using either piecewise polynomials or RBFs.
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Details
Title
- Optimal sampling for linear function approximation and high-order finite difference methods over complex regions
Contributors
- Liu, Tony (Author)
- Platte, Rodrigo B (Thesis advisor)
- Renaut, Rosemary (Committee member)
- Kaspar, David (Committee member)
- Moustaoui, Mohamed (Committee member)
- Motsch, Sebastien (Committee member)
- Arizona State University (Publisher)
Date Created
The date the item was original created (prior to any relationship with the ASU Digital Repositories.)
2019
Subjects
Resource Type
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Note
- thesisPartial requirement for: Ph.D., Arizona State University, 2019
- bibliographyIncludes bibliographical references (pages 86-89)
- Field of study: Mathematics
Citation and reuse
Statement of Responsibility
by Tony Liu