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Effective numerical integration on complex shaped elements by discrete signed measures

Laura Rinaldi, Alvise Sommariva, Marco Vianello

TL;DR

This work develops a stable, low-cost method to compress finite measures into low-cardinality discrete signed measures using moment matching with an orthonormal polynomial basis. By employing a cheap quadrature for a bounding measure on $B$ and forming weights $\mathbf{w}=D V \mathbf{m}$ that satisfy $V^T \mathbf{w}=\mathbf{m}$, the authors achieve exact integration for polynomials up to degree $n$ without matrix conditioning issues, and they provide robust $\ell_1$-norm bounds. The approach is demonstrated on two practical fronts: cheap, stable quadrature on complex-shaped planar spline elements for high-order FEM/VEM, and compression of quasi-Monte Carlo measures on difficult 3D geometries, yielding significant reductions in node counts while maintaining high accuracy. The results support the method's potential for stable, efficient integration in complex geometries and repeated simulations, with open-source code available for reproducibility.

Abstract

We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.

Effective numerical integration on complex shaped elements by discrete signed measures

TL;DR

This work develops a stable, low-cost method to compress finite measures into low-cardinality discrete signed measures using moment matching with an orthonormal polynomial basis. By employing a cheap quadrature for a bounding measure on and forming weights that satisfy , the authors achieve exact integration for polynomials up to degree without matrix conditioning issues, and they provide robust -norm bounds. The approach is demonstrated on two practical fronts: cheap, stable quadrature on complex-shaped planar spline elements for high-order FEM/VEM, and compression of quasi-Monte Carlo measures on difficult 3D geometries, yielding significant reductions in node counts while maintaining high accuracy. The results support the method's potential for stable, efficient integration in complex geometries and repeated simulations, with open-source code available for reproducibility.

Abstract

We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.
Paper Structure (5 sections, 3 theorems, 28 equations, 6 figures, 7 tables)

This paper contains 5 sections, 3 theorems, 28 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Moment-based discretized measure compression by orthogonal polynomials
  3. Numerical examples
  4. Cheap and stable quadrature on planar spline curvilinear elements
  5. Compression of discrete measures with large support: QMC on complex 3D shapes

Key Result

Theorem 2.1

Let $\mu$ be a finite measure with support $\Omega\subseteq \mathbb{R}^d$, and $\lambda$ a finite measure with $\mathbb{P}_n$-determining support $B\subseteq \mathbb{R}^d$ (i.e., polynomials of total degree not exceeding $n$ which vanish there vanish everywhere in $\mathbb{R}^d$). Moreover, let $\{p Denote by $V\in \mathbb{R}^{M\times N}$ the Vandermonde-like matrix $V=(v_{ij})=(p_j(x_i))$, by $D=

Figures (6)

  • Figure 1: The planar curvilinear elements $\Omega_1$ and $\Omega_2$ and the nodes of a cheap formula with $\hbox{ADE}=10$. Green dots: nodes with positive weights; red dots: nodes with negative weights.
  • Figure 2: $121=11^2$ weights of the cheap rule with $\hbox{ADE}=10$ on the planar curvilinear elements $\Omega_1$ (left) and $\Omega_2$ (right), in increasing order.
  • Figure 3: Small crosses: relative quadrature errors for 100 trials of random polynomials $(c_0+c_1 x+c_2 y)^n$ on the elements $\Omega_1$ (left) and $\Omega_2$ (right). Circles: geometric mean of the relative errors. The abscissae are the ADE of the formulas.
  • Figure 4: The multivariate elements $\Omega_3$ and $\Omega_4$.
  • Figure 5: Weights in increasing order of the cheap QMC rules with $\hbox{ADE}=10$ on the 3D elements $\Omega_3$ (left) and $\Omega_4$ (right). The corresponding nodes are $11^3=1331$, much less than the hundreds of thousands of points in the original QMC rules.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2