Abstract. Centroidal Voronoi tessellations are useful for subdividing a region in Euclidean space into Voronoi regions whose generators are also the centers of mass, with respect to a prescribed density function, of the regions. Their extensions to general spaces and sets are also available; for example, tessellations of surfaces in a Euclidean space may be considered. In this paper, a precise definition of such constrained centroidal Voronoi tessellations (CCVTs) is given and a number of their properties are derived, including their characterization as minimizers of an "energy." Deterministic and probabilistic algorithms for the construction of CCVTs are presented and some analytical results for one of the algorithms are given. Computational examples are provided which serve to illustrate the high quality of CCVT point sets. Finally, CCVT point sets are applied to polynomial interpolation and numerical integration on the sphere.Key words. surface tessellations, optimal Voronoi tessellations, surface interpolation, surface quadrature, point sets on surfaces, point sets on the sphere AMS subject classifications. 65M50, 65N50, 65Y20, 65D05, 65D30 PII. S10648275013915761. Introduction. In [1, 2, 3, 4] and [8], a methodology for point placement in regions, i.e., volumes, in R N has been developed. The methodology is based on the notion of centroidal Voronoi tessellations (CVTs), which is explained in section 1.1. The ensuing methodology produces high-quality point distributions which may themselves be of interest or may be used as a basis for triangulations or Voronoi tessellations of the region. Among the advantages of the CVT methodology is that points may easily be distributed according to a prescribed nonuniform density function and the algorithms which make up the methodology are amenable to parallelization. In addition, CVTs enjoy an optimization characterization so that they themselves turn out to be useful in many applications such as image and data analysis, vector quantization, resource optimization, optimal placement of sensors and actuators for control, cell biology, territorial behavior of animals, numerical partial differential equations, meshless computing, etc.; see, e.g., [1,2,3,5,10].The basic definition of the CVT can be generalized to very broad settings that range from abstract spaces to discrete point sets [1]. The purpose of this paper is to study the CVT methodology that was developed in [1, 2, 3, 4] and [8] in the case where the point sets are constrained to lie on surfaces in R N . There are many instances in which point distributions lying on surfaces or triangulations or more general subdivisions of surfaces are needed. Just to mention a few important examples, there are

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.

During the next decade and beyond, climate system models will be challenged to resolve scales and processes that are far beyond their current scope. Each climate system component has its prototypical example of an unresolved process that may strongly influence the global climate system, ranging from eddy activity within ocean models, to ice streams within ice sheet models, to surface hydrological processes within land system models, to cloud processes within atmosphere models. These new demands will almost certainly result in the develop of multiresolution schemes that are able, at least regionally, to faithfully simulate these fine-scale processes. Spherical centroidal Voronoi tessellations (SCVTs) offer one potential path toward the development of a robust, multiresolution climate system model components. SCVTs allow for the generation of highquality Voronoi diagrams and Delaunay triangulations Responsible Editor: Eric Deleersnijder LA-UR-08-05303. T. Ringler (B) through the use of an intuitive, user-defined density function. In each of the examples provided, this method results in high-quality meshes where the quality measures are guaranteed to improve as the number of nodes is increased. Real-world examples are developed for the Greenland ice sheet and the North Atlantic ocean.Idealized examples are developed for ocean-ice shelf interaction and for regional atmospheric modeling. In addition to defining, developing, and exhibiting SCVTs, we pair this mesh generation technique with a previously developed finite-volume method. Our numerical example is based on the nonlinear, shallowwater equations spanning the entire surface of the sphere. This example is used to elucidate both the potential benefits of this multiresolution method and the challenges ahead.

The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While our long-term motivation is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, we begin with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of our approach. The multiresolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine-and coarse-mesh resolution. We explore the shallow-water system with a suite of meshes ranging from quasi-uniform resolution meshes, where grid spacing is globally uniform, to highly-variable resolution meshes, where grid spacing varies by a factor of 16 between the fine and coarse regions. We find that potential vorticity is conserved to within machine precision and total available energy is conserved to within time-truncation error. This finding holds for the full suite of meshes, ranging from quasi-uniform resolution and highly-variable resolution meshes. Using shallow-water test cases 2 and 5, we find that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This finding is consistent with results obtained by others.When these variable resolution meshes are used for the simulation of an unstable zonal jet, we find that the core features of the growing instability are largely unchanged as the variation in mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multi-resolution climate system modeling.1

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