Centroidal power diagrams, Lloyd's algorithm, and applications to optimal location problems

D. P. Bourne, Steven M. Roper

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Abstract

In this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems. The target energy can be written either in terms of atomic measures and the Wasserstein distance or in terms of weighted points and power diagrams (generalized Voronoi diagrams). The latter formulation is more suitable for computation. We show that critical points of the energy are centroidal power diagrams, which are generalizations of centroidal Voronoi tessellations, and that they can be approximated by a generalization of Lloyd's algorithm (Lloyd's algorithm is a common method for finding centroidal Voronoi tessellations). We prove that the algorithm is energy decreasing and prove a convergence theorem. Numerical experiments suggest that the algorithm converges linearly. We illustrate the algorithm in two and three dimensions using simple models of optimal location and crystallization (see online supplementary material).

Original languageEnglish
Pages (from-to)2545-2569
Number of pages25
JournalSIAM Journal on Numerical Analysis
Volume53
Issue number6
DOIs
Publication statusPublished - 5 Nov 2015

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Keywords

  • Lloyd's method
  • Optimal location problems
  • Power diagram
  • Quantization
  • Voronoi diagram

ASJC Scopus subject areas

  • Numerical Analysis

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