This paper explores the reconstruction of a real-valued function f defined over a domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varOmega \subset \mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of f over a given triangulation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {T}_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">T</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varOmega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>Ω</mi> </math></EquationSource> </InlineEquation> are available and seek a bivariate polynomial that approximates f using a histopolation approach, potentially flanked by an additional regression technique. This methodology relies on the selection of a subset of triangles <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {T}_M \subset \mathcal {T}_N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">T</mi> <mi>M</mi> </msub> <mo>⊂</mo> <msub> <mi mathvariant="script">T</mi> <mi>N</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> 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.

Bivariate polynomial histopolation techniques on Padua, Fekete, and Leja triangles

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2026年5月4日

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Ludovico Bruni Bruno,
Francesco Dell’Accio,
Wolfgang Erb,
Federico Nudo

摘要

This paper explores the reconstruction of a real-valued function f defined over a domain $\varOmega \subset \mathbb {R}^2$ $Ω \subset R^{2}$ using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of f over a given triangulation $\mathcal {T}_N$ $T_{N}$ of $\varOmega $ $Ω$ 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 $\mathcal {T}_M \subset \mathcal {T}_N$ $T_{M} \subset T_{N}$ 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.