Constructive quasi-uniform sequences over triangles

In this paper, we develop constructive algorithms for generating quasi-uniform point sets and sequences over arbitrary two-dimensional triangular domains. Our proposed method, called the \emph{Voronoi-guided greedy packing} algorithm, iteratively selects the point farthest from the current set among a finite candidate set determined by the Voronoi diagram of the triangle. Our main theoretical result shows that, after a finite number of iterations, the mesh ratio of the generated point set is at most~2, which is known to be optimal. We further analyze two existing triangular low-discrepancy point sets and prove that their mesh ratios are uniformly bounded, thereby establishing their quasi-uniformity. Finally, through a series of numerical experiments, we demonstrate that the proposed method provides an efficient and practical strategy for generating high-quality point sets on individual triangles.

• Publication year

  2025

• Venue

  arXiv.org

• Publication date

  2025-11-11

• Fields of study

  Mathematics, Computer Science

• Identifiers
  DOI 10.48550/arXiv.2511.07909arXiv 2511.07909
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part I: Lattices and Kronecker sequences

  2025cited by this paper
• On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences

  2025cited by this paper
• One-dimensional quasi-uniform Kronecker sequences

  2024cited by this paper
• Randomized Quasi-Monte Carlo Methods on Triangles: Extensible Lattices and Sequences

  2024cited by this paper
• The Sobol' sequence is not quasi-uniform in dimension 2

  2023cited by this paper
• Quasi-uniform designs with optimal and near-optimal uniformity constant

  2021cited by this paper
• A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability & uniform point distribution

  2019cited by this paper
• Quasi-Monte Carlo integration for twice differentiable functions over a triangle

  2017cited by this paper
• Scattered Data Approximation

  2016cited by this paper
• Space-filling designs for computer experiments: A review

  2016cited by this paper
• A Comparison of Popular Point Configurations on S^2

  2016cited by this paper
• Low Discrepancy Constructions in the Triangle

  2014influential reference
• Design of computer experiments: space filling and beyond

  2011cited by this paper
• Stability of kernel-based interpolation

  2010cited by this paper
• Indexing the Sphere with the Hierarchical Triangular Mesh

  2007cited by this paper
• A spatial data structure for fast Poisson-disk sample generation

  2006cited by this paper
• Kernel techniques: From machine learning to meshless methods

  2006cited by this paper
• Design and Modeling for Computer Experiments

  2005cited by this paper
• A numerical study of some radial basis function based solution methods for elliptic PDEs

  2003cited by this paper
• The finite element method for elliptic problems

  2002cited by this paper
• A General Framework for Mesh Decimation

  1998cited by this paper
• Design and analysis of computer experiments

  1998cited by this paper
• Error estimates and condition numbers for radial basis function interpolation

  1995cited by this paper
• Decimation of triangle meshes

  1992cited by this paper
• Random number generation and Quasi-Monte Carlo methods

  1992cited by this paper
• Minimax and maximin distance designs

  1990cited by this paper
• Deterministic and Stochastic Error Bounds in Numerical Analysis

  1988influential reference
• A sweepline algorithm for Voronoi diagrams

  1986cited by this paper
• ON THE ANGLE CONDITION IN THE FINITE ELEMENT METHOD

  1976cited by this paper
• Uniform distribution of sequences

  1974cited by this paper

• No citing papers are available for this paper.