Some Problems on Analyses and Applications of Centroidal Voronoi Tessellations

质心Voronoi镶嵌（cvt）是一种特殊的Voronoi镶嵌，具有Voronoi镶嵌的生成器也是相应Voronoi细胞的给定密度函数的质心的性质。cvt在图像和数据分析、矢量量化、计算机图形学、资源优化、细胞生物学、偏微分方程数值解、最优控制、移动传感网络等研究领域都有广泛的应用。本项目旨在进一步分析cvt的性质和算法，并拓宽cvt和相关概念的应用范围，为更有效和准确的治疗奠定基础。在理论方面，将研究计算cvt的常用算法的收敛和加速方案，以及在其他度量设置和具有层次结构的cvt的泛化。在应用方面，研究者将开发和实现鲁棒的基于cvt的网格生成和优化算法，然后将其结合到使用有限元方法或有限体积方法的数值偏微分方程的自适应计算中，以及在球面和其他表面（如地球物理流动）上解决一些具有挑战性的物理问题。此外，还将考虑基于CVT方法的皮质表面平坦化技术，这对脑成像数据分析非常重要，包括功能变异性的定量映射和概率脑表面地图集的构建。所提出的研究将为许多突出的理论问题提供新的见解，并导致科学和工程中各种重要应用的计算算法的革新。这个项目产生的软件将积极地传播，这样它不仅可以被科学计算领域的研究人员使用，而且可以被更广泛的社区的实践者应用于跨学科科学的问题。

Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations having the property that the generators of the Voronoi tessellation are also the centers of mass, with respect to a given density function, of the corresponding Voronoi cells. CVTs have been very usedful in a wide range of research fields, including image and data analysis, vector quantization, computer graphics, resource optimization, cell biology, numerical solution of partial differential equations, optimal control, mobile sensing networks, and so on. This project aims at further analysis on CVTs' properties and algorithms, and the broadening of applications for which CVTs and related concepts can be used as a basis for more efficient and accurate treatments.In the theoretical aspects, the convergence and acceleration schemes of popular algorithms for computing CVTs and generalization of CVTs in other metric settings and with hierarchical structures will be studied. In the application aspects, the investigator will develop and implement robust CVT-based mesh generation and optimization algorithms, and then incorporate them in adaptive computations of numerical partial differential equations using finite element methods or finite volume methods, and in the solution of some challenging physical problems on the sphere and other surfaces such as geophysical flows, in light of the high-quality CVT-based surface meshing.Also considered will be the cortical surface-flattening techniques based on the CVT methodology, which are very important to brain-imaging data analysis, including quantitative mapping of functional variability and construction of probabilistic brain-surface atlases. The proposed research will offer new insight into a number of outstanding theoretical issues and lead to renovation of computational algorithms for diverse important applications in science and engineering. The software resulting from this project will be actively disseminated, so that it can be used not only by researchers in the scientific computing area, but also by practitioners in a much broader community for application to problems in interdisciplinary sciences.