Probabilistic and parallel algorithms for centroidal Voronoi tessellations with application to meshless computing and numerical analysis on surfaces

Abstract

Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, and grid generation. A detailed review is given in Chapter 1. In Chapter 2, some probabilistic methods for determining centroidal Voronoi tessel­ lations and their parallel implementation on distributed memory systems are presented. The results of computational experiments performed on a CRAY T3E-600 system are given for each algorithm. These demonstrate the superior sequential and parallel per­ formance of a new algorithm we introduce. Then, new algorithms are presented in Chapter 3 for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or su­ perimposed meshes. Computational examples are provided that show, for both uniform and nonuniform point distributions, that the algorithms result in high-quality point sets and high-quality support regions. The extensions of centroidal Voronoi tessellations to general spaces and sets are also available. For example, tessellations of surfaces in an Euclidean space may be considered. In Chapter 4, a precise definition of such constrained centroidal Voronoi tessellations (CCVT's) is given and a number of their properties are derived, including