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，其性质是Voronoi镶嵌的生成元相对于给定的密度函数也是相应的Voronoi网格的质量中心。连续小波变换在图像与数据分析、矢量量化、计算机图形学、资源优化、细胞生物学、偏微分方程组数值求解、最优控制、移动传感网络等领域有着广泛的应用。本项目旨在进一步分析CVT的性质和算法，拓宽CVT及其相关概念的应用范围，以便更有效和准确地处理CVT。在理论方面，将研究计算CVT的流行算法的收敛和加速方案，以及CVT在其他度量设置和分层结构下的推广。在应用方面，研究人员将开发和实现稳健的基于CVT的网格生成和优化算法，然后将其应用于使用有限元方法或有限体积方法的数值偏微分方程组的自适应计算，以及针对球和其他表面上的一些具有挑战性的物理问题的解决方案，如高质量的基于CVT的表面网格划分。此外，还将考虑基于CVT方法的皮质表面平坦化技术，这些技术对于脑成像数据分析非常重要，包括定量绘制功能变异性图和构建概率脑表面图谱。拟议的研究将为一些悬而未决的理论问题提供新的见解，并导致计算算法的更新，用于科学和工程中的各种重要应用。该项目产生的软件将得到积极传播，使其不仅可供科学计算领域的研究人员使用，而且还可供更广泛社区的从业人员应用于跨学科科学问题。

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.