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

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

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).

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Roper, Dr Steven and Bourne, Dr David
• Authors: Bourne, D.P., and Roper, S.M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: SIAM: Journal on Numerical Analysis
• Publisher: Society for Industrial and Applied Mathematics
• ISSN: 0036-1429
• ISSN (Online): 1095-7170
• Published Online: 05 November 2015
• Copyright Holders: Copyright © 2015 Society for Industrial and Applied Mathematics
• First Published: First published in SIAM: Journal on Numerical Analysis 53(6): 2545-2569
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.