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Piecewise Padé-Chebyshev Reconstruction of Bivariate Piecewise Smooth Functions

Akansha Singh

TL;DR

This work addresses the problem of approximating piecewise smooth functions in one and two dimensions while mitigating the Gibbs phenomenon, without requiring prior knowledge of singularity locations. It develops PiPC for univariate cases and Pi2DPC for the bivariate case, both built on Maehly's Padé-Chebyshev framework and realized in a piecewise, local fashion. Key contributions include a robust framework for computing Padé-Chebyshev approximants in 1D and 2D (with piecewise variants) that yields linear systems with Toeplitz/Hankel structure, and numerical demonstrations showing substantially reduced Gibbs oscillations, particularly for functions with axis-aligned singularities. The approach offers a flexible, knowledge-free tool for high-fidelity multivariate approximation with potential applications in PDEs and physics, outperforming global Chebyshev and non-piecewise Padé methods in capturing singularities.

Abstract

We extend the idea of approximating piecewise smooth univariate functions using rational approximation introduced in \cite{aka_bas-19a} to two-dimensional space. This article aims to implement the novel piecewise Maehly based Padé-Chebyshev approximation \cite{mae_60a}. We first develop a method referred to as PiPC to approximate univariate piecewise smooth functions and then extend the same to a two-dimensional space, leading to a bivariate piecewise Padé-Chebyshev approximation (Pi2DPC) for approximating piecewise smooth functions in two-dimension. We study the utility of the proposed techniques in minimizing the Gibbs phenomenon while approximating piecewise smooth functions. The chief advantage of these methods lies in their non-dependence on any apriori knowledge of the locations and types of singularities (if any) present in the original function. Finally, we supplement our methods with numerical results to validate their effectiveness in diminishing the Gibbs phenomenon to negligible levels.

Piecewise Padé-Chebyshev Reconstruction of Bivariate Piecewise Smooth Functions

TL;DR

This work addresses the problem of approximating piecewise smooth functions in one and two dimensions while mitigating the Gibbs phenomenon, without requiring prior knowledge of singularity locations. It develops PiPC for univariate cases and Pi2DPC for the bivariate case, both built on Maehly's Padé-Chebyshev framework and realized in a piecewise, local fashion. Key contributions include a robust framework for computing Padé-Chebyshev approximants in 1D and 2D (with piecewise variants) that yields linear systems with Toeplitz/Hankel structure, and numerical demonstrations showing substantially reduced Gibbs oscillations, particularly for functions with axis-aligned singularities. The approach offers a flexible, knowledge-free tool for high-fidelity multivariate approximation with potential applications in PDEs and physics, outperforming global Chebyshev and non-piecewise Padé methods in capturing singularities.

Abstract

We extend the idea of approximating piecewise smooth univariate functions using rational approximation introduced in \cite{aka_bas-19a} to two-dimensional space. This article aims to implement the novel piecewise Maehly based Padé-Chebyshev approximation \cite{mae_60a}. We first develop a method referred to as PiPC to approximate univariate piecewise smooth functions and then extend the same to a two-dimensional space, leading to a bivariate piecewise Padé-Chebyshev approximation (Pi2DPC) for approximating piecewise smooth functions in two-dimension. We study the utility of the proposed techniques in minimizing the Gibbs phenomenon while approximating piecewise smooth functions. The chief advantage of these methods lies in their non-dependence on any apriori knowledge of the locations and types of singularities (if any) present in the original function. Finally, we supplement our methods with numerical results to validate their effectiveness in diminishing the Gibbs phenomenon to negligible levels.

Paper Structure

This paper contains 9 sections, 1 theorem, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Univariate Chebyshev series expansion
  3. Univariate Padé-Chebyshev Reconstruction
  4. Bivariate Chebyshev Series Expansion
  5. Construction of Piecewise bivariate Chebyshev Approximation
  6. Bivariate Padé-Chebyshev Approximation
  7. Piecewise 2D Padé-Chebyshev Approximation
  8. Numerical Comparison
  9. Conclusion

Key Result

Theorem 4.1

(Mason mas-han-03a) Let $f(x,y)$ be a continuous function of bounded variation with one of the partial derivative is bounded in $D$. Then $f$ can be represented by the Chebyshev series as

Figures (5)

  • Figure 1: Depicts the function $f(x,y)= sign(4xy)$ on the unit square $D=[-1,1]^2$.
  • Figure 2: Depicts (a) global bivariate Chebyshev, (b) global bivariate Padé-Chebyshev, (c) Pi2DC and (d) Pi2DPC approximants of the function $f(x,y)= sign(4xy), (x,y)\in D$.
  • Figure 3: The corresponding pointwise error plots are depicted with $L^\infty$-errors.
  • Figure 4: (a) Depicts the surface plot of the function defined in \ref{['eq:disc2']} on the unit square $[-1,1]^2$. (b) depicts Pi2DC approximant, and (c) depicts Pi2DPC approximant of $h(x,y)$.
  • Figure 5: Comparision of the slice of the surface of Pi2DC and Pi2DPC approximants of $h(x,y)$ along with the zoomed plot.

Theorems & Definitions (3)

  • Theorem 4.1
  • Example 6.1
  • Example 6.2