Piecewise Pad\'e-Chebyshev Reconstruction of Bivariate Piecewise Smooth Functions

TL;DR

This paper introduces a novel two-dimensional piecewise Padé-Chebyshev approximation method that effectively reduces the Gibbs phenomenon in approximating piecewise smooth functions without prior knowledge of singularities.

Contribution

It extends univariate rational approximation techniques to 2D, developing the Pi2DPC method for improved approximation of piecewise smooth functions.

Findings

Successfully minimizes Gibbs phenomenon in numerical experiments

Does not require prior knowledge of singularity locations or types

Demonstrates effectiveness in two-dimensional function approximation

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 Pad\'e-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 Pad\'e-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.