Bivariate polynomial histopolation techniques on Padua, Fekete and Leja triangles

TL;DR

This paper develops and evaluates bivariate polynomial histopolation methods on specialized node sets like Padua points, Fekete, and Leja triangles, for reconstructing functions from average values over triangulations.

Contribution

It introduces combined histopolation-regression techniques on these node sets, ensuring solvability and improved approximation accuracy.

Findings

Algorithms are successfully implemented and tested.

Methods demonstrate effective function approximation.

Use of specialized nodes enhances stability and accuracy.

Abstract

This paper explores the reconstruction of a real-valued function $f$ defined over a domain $\Omega \subset \mathbb{R}^2$ using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of $f$ over a given triangulation $\mathcal{T}_N$ of $\Omega$ are available and seek a bivariate polynomial that approximates $f$ using a histopolation approach, potentially flanked by an additional regression technique. This methodology relies on the selection of a subset of triangles $\mathcal{T}_M \subset \mathcal{T}_N$ for histopolation, ensuring both the solvability and the well-conditioning of the problem. The remaining triangles can potentially be used to enhance the accuracy of the polynomial approximation through a simultaneous regression. We will introduce histopolation and combined histopolation-regression methods using the Padua points, discrete Leja sequences, and approximate Fekete nodes. The proposed algorithms are implemented and evaluated through numerical experiments that demonstrate their effectiveness in function approximation.