Calculus of Variations and Geometric Measure Theory
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D. Bourne - S. Roper

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

created by bourne on 09 Sep 2014
modified on 13 Jan 2016


Published Paper

Inserted: 9 sep 2014
Last Updated: 13 jan 2016

Journal: SIAM Journal on Numerical Analysis
Volume: 53
Number: 6
Pages: 2545-2569
Year: 2015
Doi: 10.1137/141000993


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).


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