Study on Algorithms and Applications of Centroidal Voronoi Tessellations

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This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations having the propertythat the generators of the Voronoi tessellations are also the centroids, with respect to a given density function, of the corresponding Voronoi cells. In this project, we will continue to investigate algorithms for computing CVTs and CVT-based applications for scientific and engineering problems. Topics of the proposed project include: study of single limit-point convergence analysis for the Lloyd's algorithm; development and analysis of nonlinear conjugate gradient methods for computing CVTs; study and implementation of parallel CVT/CVDT mesh generation on the distributed systems; improving existing CCVT-based techniques for surface meshing; incorporating these meshing schemes in adaptive solutions of partial differential equations, especially for the convection-dominated problems; and further investigation and improvement of the edge-weighted CVT model and corresponding algorithms for image segmentation that combines the intensity information in the color space of the image and the local edge information in the physical space. CVT-based methodologies have been proven to be very useful in diverse applications in the past decade, including but not limited to, image processing, vector quantization and data analysis, resource optimization, optimal placement of sensors and actuators for control, cell biology and territorial behavior of animals, high-quality point sampling, mesh generation and optimization, numerical partial differential equations, climate and atmospheric science, model reduction, computer graphics and vision, mobile sensing networks, logistics system design, and etc. The application list is still growing. The proposed project has a comprehensive coverage of algorithm design and analysis, implementation and applications of CVTs to diverse problems in science and engineering. Mathematical tools are used to analyze these techniques to give guidelines for their applicability; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in real applications and large scale computations. The proposed investigation will offer new insight into the understanding of the elegant Lloyd's algorithm and it will also lead to exploration of transformative concepts and renovation of computational algorithms for many important applications involving mesh optimization, adaptive algorithms, energy minimizationand image processing based on the CVT methodologies. In addition, this project will also offer a unique educational opportunity for graduate students with interests in computational and applied mathematics, engineering and information technology by having them participate in an interdisciplinary research program.