Abstract. Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi tessellations may also be defined in more abstract and more general settings. Due to the natural optimization properties enjoyed by CVTs, they have many applications in diverse fields. The Lloyd algorithm is one of the most popular iterative schemes for computing the CVTs but its theoretical analysis is far from complete. In this paper, some new analytical results on the local and global convergence of the Lloyd algorithm are presented. These results are derived through careful utilization of the optimization properties shared by CVTs. Numerical experiments are also provided to substantiate the theoretical analysis.Key words. centroidal Voronoi tessellations, k-means, optimal vector quantizer, Lloyd algorithm, global convergence, convergence rate AMS subject classifications. 65D99, 65C20 DOI. 10.1137/040617364 1. Introduction. A centroidal Voronoi tessellation (CVT) is a special Voronoi tessellation of a given set such that the associated generating points are the centroids (centers of mass) of the corresponding Voronoi regions with respect to a predefined density function [7]. CVTs are indeed special as they enjoy very natural optimization properties which make them very popular in diverse scientific and engineering applications that include art design, astronomy, clustering, geometric modeling, image and data analysis, resource optimization, quadrature design, sensor networks, and numerical solution of partial differential equations [1,2,3,4,7,8,9,10,11,13,14,17,15,26,29,30,31,39,44,45]. In particular, CVTs have been widely used in the design of optimal vector quantizers in electrical engineering [25,28,40,43]. They are also related to the so-called method of k-means [27] in clustering analysis. CVTs can also be defined in more general cases such as those constrained to a manifold [12,11] or those corresponding to anisotropic metrics [16,18], and other abstract settings [7,9].For modern applications of the CVT concept in large-scale scientific and engineering problems, it is important to develop robust and efficient algorithms for constructing CVTs in various settings. Historically, a number of algorithms have been studied and widely used [7,19,25,27,38]. A seminal work is the algorithm first developed in the 1960s at Bell Laboratories by S. Lloyd which remains to this day one of the most popular methods due to its effectiveness and simplicity. The algorithm was later officially published in [35]. It is now commonly referred to as the Lloyd algorithm and is the main focus of this paper.

Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in polycrystals. The grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the growth process, an initially random distribution of boundary types reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, it is found that if the given energy density depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy.In this paper we derive an entropy based theory which suggests that the evolution of the GBCD satisfies a Fokker-Planck Equation, an equation whose stationary state is a Boltzmann distribution. Cellular structures coarsen according to a local evolution law, curvature driven growth, and are limited by space filling constraints. The interaction between the evolution law and the constraints is governed primarily by the force balance at triple junctions, the natural boundary condition associated to curvature driven growth, and determines a dissipation relation. A simplified coarsening model is introduced which is driven by the boundary conditions and reflects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free springmass-dashpots. Application is made of the recent characterization of Fokker-Planck kinetics as a gradient flow for a free energy in deriving the theory. The theory predicts the results of large scale 2D simulations and is consistent with experiment.

Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains-a texture-appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fokker-Planck Equation

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SUMMARYCentroidal Voronoi tessellations (CVT) have diverse applications in many areas of science and engineering. The development of e cient algorithms for their construction is a key to their success in practice. In this paper, we study some new algorithms for the numerical computation of the CVT, including the Lloyd-Newton iteration and the optimization based multilevel method. Both theoretical analysis and computational simulations are conducted. Rigorous convergence results are presented and signiÿcant speedup in computation is demonstrated through the comparison with traditional methods.