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ORTHOCUB: integral and differential cubature rules by orthogonal moments

Laura Rinaldi, Alvise Sommariva, Marco Vianello

TL;DR

ORTHOCUB provides a matrix-free framework to approximate linear functionals on multivariate polynomials by projecting onto an orthonormal basis with moments, using near-minimal algebraic cubature rules on a bounding box. The method unifies integration and differentiation functionals under a hyperinterpolation perspective, enabling exact polynomial recovery up to degree n without solving linear systems. By specializing to box domains and employing MPX product Chebyshev rules, ORTHOCUB achieves significant reductions in node counts while maintaining stability, with demonstrated efficiency in FEM/VEM contexts and QMC data compression. The written software in Matlab/Python, plus a set of demos, validates accuracy, stability, and practical usability for high-dimensional cubature tasks and differential operators on complex geometries.

Abstract

We discuss a numerical package, named ORTHOCUB, for the computation of linear functionals of both integral and differential type on multivariate polynomial spaces. The weighted sums corresponding to such integral and differential cubatures are implemented via orthogonal polynomial moments and auxiliary near-minimal algebraic cubature in a bounding box, with no conditioning issue since no matrix inversion or factorization is needed. The whole computational process indeed reduces to moment computation and dense matrix-vector products of relatively small size. The Matlab and Python codes are freely available, to be used as building blocks for integral and differential problems.

ORTHOCUB: integral and differential cubature rules by orthogonal moments

TL;DR

ORTHOCUB provides a matrix-free framework to approximate linear functionals on multivariate polynomials by projecting onto an orthonormal basis with moments, using near-minimal algebraic cubature rules on a bounding box. The method unifies integration and differentiation functionals under a hyperinterpolation perspective, enabling exact polynomial recovery up to degree n without solving linear systems. By specializing to box domains and employing MPX product Chebyshev rules, ORTHOCUB achieves significant reductions in node counts while maintaining stability, with demonstrated efficiency in FEM/VEM contexts and QMC data compression. The written software in Matlab/Python, plus a set of demos, validates accuracy, stability, and practical usability for high-dimensional cubature tasks and differential operators on complex geometries.

Abstract

We discuss a numerical package, named ORTHOCUB, for the computation of linear functionals of both integral and differential type on multivariate polynomial spaces. The weighted sums corresponding to such integral and differential cubatures are implemented via orthogonal polynomial moments and auxiliary near-minimal algebraic cubature in a bounding box, with no conditioning issue since no matrix inversion or factorization is needed. The whole computational process indeed reduces to moment computation and dense matrix-vector products of relatively small size. The Matlab and Python codes are freely available, to be used as building blocks for integral and differential problems.

Paper Structure

This paper contains 11 sections, 1 theorem, 43 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Weighted sum functionals by orthogonal moments
  3. Implementation
  4. Near-minimal rules for the product Chebyshev measure
  5. ORTHOCUB and Hyperinterpolation
  6. Applications and demos
  7. Integral cubature
  8. Differential cubature
  9. Software
  10. Description of the main codes
  11. Description of the demos

Key Result

Theorem 2.1

Let $B$ be (the closure of) an open set in $\mathbb{R}^d$ and $d\mu=\sigma(P)\,dP$ an absolutely continuous measure with respect to the Lebesgue measure on $B$. Moreover, let $\{\phi_j\}_{1\leq j\leq N}$ be a $\mu$-orthonormal basis for $\mathbb{P}_n$ (the polynomials of total-degree not exceeding $ Let $\mathcal{L}: S\to \mathbb{R}$ a linear functional, where $S$ is a subspace of real-valued func

Figures (5)

  • Figure 1: ORTHOCUB integration on a curved element with spline boundary (green dots: nodes with positive weights; red dots: nodes with negative weights): sign and size distribution of the weights for ADE $n=10$, relative integration errors (crosses) for 100 trials of random polynomials $p_n(x,y)=(c_0+c_1 x+c_2 y)^n$, $n=2,4,\dots,16$, with their geometric mean (circles).
  • Figure 2: ORTHOCUB compression of QMC integration on the union of 5 balls: sign and size distribution of the weights for ADE $n=10$, relative integration errors (crosses) for 100 trials of random polynomials $p_n(x,y,z)=(c_0+c_1 x+c_2 y+c_3z)^n$, $n=2,4,\dots,16$, with their geometric mean (circles).
  • Figure 3: Small crosses: relative differentiation errors in the 2-norm on the first 100 Halton points in $[-1,1]^2$, for 100 trials of random polynomials $p_n(x,y)=(c_0+c_1 x+c_2 y)^n$, $n=2,4,\dots,16$. Circles: geometric mean of the relative errors.
  • Figure 4: As in Figure \ref{['partial2D']} for the random polynomials $p_n(x,y,z)=(c_0+c_1 x+c_2 y+c_3z)^n$ in $[-1,1]^3$.
  • Figure 5: The "Lebesgue constants" (\ref{['lebdiff']}) of differential cubature in $[-1,1]^2$ (red) and $[-1,1]^3$ (blue) with polynomial degree $n=2,4,\dots,16$: top: first partial derivatives (crosses), with their quadratic least-square fitting (circles); bottom: mixed (left) and second (right) partial derivatives (crosses), with their quartic least-square fitting (circles).

Theorems & Definitions (3)

  • Theorem 2.1
  • Remark 4.1
  • Remark 4.2