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

Bourne, D.P.; Roper, S.M.

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).

Citation

Bourne, D., & Roper, S. (2015). Centroidal power diagrams, Lloyd's algorithm and applications to optimal location problems. SIAM Journal on Numerical Analysis, 53(6), 2545-2569. https://doi.org/10.1137/141000993

• Journal Article Type: Article
• Acceptance Date: Aug 24, 2015
• Online Publication Date: Nov 5, 2015
• Publication Date: Nov 5, 2015
• Deposit Date: Dec 22, 2015
• Publicly Available Date: Jan 19, 2016
• Journal: SIAM Journal on Numerical Analysis
• Print ISSN: 0036-1429
• Electronic ISSN: 1095-7170
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 6
• Pages: 2545-2569
• DOI: https://doi.org/10.1137/141000993
• Keywords: Voronoi diagram, Power diagram, Lloyd's method, Optimal location problems, Quantization.
• Public URL: https://durham-repository.worktribe.com/output/1424408