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Centroidal Voronoi Tessellations as Electrostatic Equilibria: A Generalized Thomson Problem in Convex Domains

Zachary Mullaghy

TL;DR

This work presents a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional, and shows that the resulting equilibrium configurations converge to CVT structures.

Abstract

We present a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional. By modeling interior point distributions in a convex domain as repelling charges balanced against a continuous boundary charge, we show that the resulting equilibrium configurations converge to CVT structures. We prove this by showing that CVTs minimize both the classical centroidal energy and the electrostatic potential, establishing a connection between geometric quantization and potential theory. Finally, we introduce a thermodynamic annealing scheme for global CVT optimization, rooted in Boltzmann statistics and random walk dynamics. By introducing a scheme for varying time steps (faster or slower cooling) we show that the set of minima of the centroid energy functional (and therefore the electrostatic potential) can be recovered. By recovering a set of generator locations corresponding to each minimum we can create a lattice continuation that allows for a customizable framework for individual minimum seeking.

Centroidal Voronoi Tessellations as Electrostatic Equilibria: A Generalized Thomson Problem in Convex Domains

TL;DR

This work presents a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional, and shows that the resulting equilibrium configurations converge to CVT structures.

Abstract

We present a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional. By modeling interior point distributions in a convex domain as repelling charges balanced against a continuous boundary charge, we show that the resulting equilibrium configurations converge to CVT structures. We prove this by showing that CVTs minimize both the classical centroidal energy and the electrostatic potential, establishing a connection between geometric quantization and potential theory. Finally, we introduce a thermodynamic annealing scheme for global CVT optimization, rooted in Boltzmann statistics and random walk dynamics. By introducing a scheme for varying time steps (faster or slower cooling) we show that the set of minima of the centroid energy functional (and therefore the electrostatic potential) can be recovered. By recovering a set of generator locations corresponding to each minimum we can create a lattice continuation that allows for a customizable framework for individual minimum seeking.

Paper Structure

This paper contains 17 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation: Electrostatics with Boundary Balancing
  3. Centroidal Voronoi Tessellations and Energy Functionals
  4. Variational Equivalence of Centroid Energy Functionals
  5. CVTs as Local Minima of Electrostatic Potential via Edge-Based Energy
  6. Hard Boundary Constraints via Infinite Potential
  7. Thermodynamic Annealing and Random Walks for Global Optimization
  8. Boltzmann Distribution over Generator Configurations
  9. Annealing Dynamics via Random Walk Perturbations
  10. Cooling Schedules and Convergence
  11. Physical Interpretation and Entropy Flow
  12. Lattice Anchoring and Variable Annealing for Metastable CVT Mapping
  13. Energy Depth and Annealing Timescale
  14. Lattice Anchoring of Local Minima
  15. LAAM Method
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\{p_i\}_{i=1}^n \subset \Omega \subset \mathbb{R}^2$ be a Centroidal Voronoi Tessellation (CVT) with corresponding Voronoi cells $V_i$. Define: where $\ell_{ij}$ is the length of the shared Voronoi edge between $V_i$ and $V_j$. Then in a neighborhood of the CVT configuration, the second variations of the two functionals are related by: for all perturbations $\delta p$, with $\lambda(\theta)

Theorems & Definitions (13)

  • Remark
  • Definition 1: Centroidal Voronoi Tessellation (CVT)
  • Definition 2: Centroid Energy Functional
  • Theorem 1: Local Equivalence of Centroidal Energy Functionals
  • proof
  • Remark
  • Definition 3: Electrostatic Potential Functional
  • Theorem 2: CVTs Minimize Electrostatic Potential Locally
  • proof
  • Remark
  • ...and 3 more