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

TL;DR

This paper introduces a numerical method based on generalized Lloyd's algorithm for solving optimal location problems using centroidal power diagrams, with proven convergence and applications in materials science.

Contribution

It develops a new computational approach for optimal location problems using centroidal power diagrams and proves its convergence, extending Lloyd's algorithm.

Findings

Algorithm converges linearly in numerical experiments.

Critical points are characterized as centroidal power diagrams.

Application to BCC lattice optimality conjecture.

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. In particular, we test a conjecture about the optimality of the BCC lattice for a simplified model of block copolymers.