Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method

Authors:
Anne Gelb
Affiliations:
Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287-1804. ag@math.la.asu.edu
Venue:
Journal of Scientific Computing
Year:
2004

Citing 6
Cited 9

On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function

Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
On the Gibbs Phenomenon and Its Resolution

SIAM Review
Determination of the jumps of a bounded function by its Fourier series

Journal of Approximation Theory
Detection of Edges in Spectral Data II. Nonlinear Enhancement

SIAM Journal on Numerical Analysis
Reducing the Effects of Noise in Image Reconstruction

Journal of Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,

Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,

Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon

Journal of Computational and Applied Mathematics
Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations

Journal of Computational Physics
Short note: Padé-Gegenbauer suppression of Runge phenomenon in the diagonal limit of Gegenbauer approximations

Journal of Computational Physics
Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Journal of Scientific Computing
Short Note: On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon

Journal of Computational Physics
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems

Journal of Scientific Computing
Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection

Journal of Scientific Computing
Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices

Journal of Computational and Applied Mathematics
Optimal Gegenbauer quadrature over arbitrary integration nodes

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.02

Visualization

Abstract

The Gegenbauer reconstruction method has been successfully implemented to reconstruct piecewise smooth functions by both reducing the effects of the Gibbs phenomenon and maintaining high resolution in its approximation. However, it has been noticed in some applications that the method fails to converge. This paper shows that the lack of convergence results from both poor choices of the parameters associated with the method, as well as numerical round off error. The Gegenbauer polynomials can have very large amplitudes, particularly near the endpoints x=±1, and hence the approximation requires that the corresponding computed Gegenbauer coefficients be extremely small to obtain spectral convergence. As is demonstrated here, numerical round off error interferes with the ability of the computed coefficients to decay properly, and hence affects the method's overall convergence. This paper addresses both parameter optimization and reduction of the round off error for the Gegenbauer reconstruction method, and constructs a viable “black box” method for choosing parameters that guarantee both theoretical and numerical convergence, even at the jump discontinuities. Validation of the Gegenbauer reconstruction method through a-posteriori estimates is also provided.