Ordered Distributions and Wavelets on Two-Dimensional Manifolds

这个项目有两个主要目标：(1)确定欧氏空间中限制在流形上的点的最小能量构形的渐近行为；(2)在流形上构造和应用小波。这个项目主要关注二维流形(表面)，因为它们在计算机图形学、生物膜和材料科学中的应用非常重要。将研究曲面上最小能量构型的几何和解析性质与该曲面的几何性质之间的关系，并开发快速生成曲面上均匀分布点集的算法。这个项目的第二个目标是发展一类新的小波理论，这些小波由“可细化的宏元素”生成，用于有效地表示曲面和曲面上的数据。该项目将研究几何建模、计算机图形学和多尺度方法在科学计算中的应用。该项目的主要目标是开发有效的方法来离散表示二维曲面和定义在这些曲面上的数据。小波和多分辨率分析领域的最新发展为有效地表示科学和工程中出现的大类数据提供了工具和统一的框架。该项目的第一个组成部分的目标是发展一类曲面上的非均匀小波理论，并开发计算机图形学和科学计算的多尺度高性能应用程序。该项目第二部分的目标是研究分布在表面上并通过成对排斥相互作用而相互作用的大量点的最小能量(或“基态”)构型的几何和分析性质。最小能量点及其在离散化流形上的应用研究将对数据采样、最优布局问题、几何设计等方法具有重要意义。均匀分布点的快速生成算法的发展在计算复杂性理论中具有重要意义。此外，对曲面上粒子基态构型有序性的研究将有助于加深对膜和薄膜物理的理解。

This project has two primary goals: (1) the determination of the asymptotic behavior of minimum energy configurations of points restricted to a manifold in Euclidean space, and (2) the construction and application of wavelets on manifolds. This project focuses on two-dimensional manifolds (surfaces) because of their importance in applications to computer graphics, biological membranes, and materials science. The connection between geometrical and analytical properties of minimum energy configurations on a surface and the geometrical properties of that surface will be investigated and algorithms for the rapid generation of well-distributed point sets on surfaces will be developed. The second goal of this project is to develop the theory of a new class of wavelets generated from "refinable macroelements" for the efficient representation of surfaces and data on surfaces. Applications to geometric modeling, computer graphics, and multiscale methods in scientific computing will be investigated.The main objective of this project is to develop effective methods for discrete representations of two-dimensional surfaces and data defined on these surfaces. The recent development of the field of wavelets and multiresolution analysis has provided tools and a unifying framework for efficiently representing large classes of data arising in science and engineering. The goals of the first component of this project are to develop the theory of a class of "nonuniform" wavelets on surfaces and to develop multiscale high-performance applications to computer graphics and scientific computing. The goal of the second component of this project is the investigation of geometrical and analytical properties of minimum energy (or "ground state") configurations of large numbers of points distributed on a surface and interacting via a pairwise repulsive interaction. The research on minimum energy points and its usefulness in discretizing manifolds will be of significance to methods for data sampling, best-packing problems, and geometric design. The development of fast algorithms for generating uniformly distributed points is of significance in computational complexity theory. Furthermore, the investigation of the ordering of ground state configurations of particles on curved surfaces will improve understanding of the physics of membranes and films.