2023-12-07T06:46:50Zhttps://keep.lib.asu.edu/oai/requestoai:keep.lib.asu.edu:node-1576492021-08-27T02:47:01Zoai_pmh:alloai_pmh:repo_items157649
https://hdl.handle.net/2286/R.I.54897
http://rightsstatements.org/vocab/InC/1.0/
2019
viii, 89 pages : illustrations (some color)
Doctoral Dissertation
Academic theses
Text
eng
Liu, Tony
Platte, Rodrigo B
Renaut, Rosemary
Kaspar, David
Moustaoui, Mohamed
Motsch, Sebastien
Arizona State University
Partial requirement for: Ph.D., Arizona State University, 2019
Includes bibliographical references (pages 86-89)
Field of study: Mathematics
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.
Applied Mathematics
Finite Difference Methods
Function Approximation
Lebesgue constant
Optimal Sampling
Sampling (Statistics)
Linear models (Statistics)
Approximation Theory
Finite differences
Polynomials
Interpolation
Functions of real variables
Optimal sampling for linear function approximation and high-order finite difference methods over complex regions