Scattered Data Analysis and Synthesis via Radial Basis Functions and Tight Spherical Frames

该项目的总体目标是开发新的方法和工具，用于通过径向基函数，球面基函数和紧球面框架分析和合成分散数据。 这项研究将提供一个基础的各种径向基函数无网格方法求解偏微分方程的有效性，估计在使用它们的错误，知道这些错误，然后可以作为一个指导，在创建和实施算法的基础上径向基函数和球面基函数，无论是无网格方法求解微分方程和离散数据曲面拟合。 紧的，局部化良好的球面框架是一种新发现的工具，用于处理球面上的分散数据，并与小波分析中使用的框架类似。 关于这些紧密框架的细节知之甚少，这个项目的第二个目标是开发它们，以及基于它们的算法，以有效地处理球体上的分散数据。涉及分析和合成从分散站点获取的数据的问题出现在不同的领域-计算机辅助设计图形，数据挖掘，医学成像，学习网络，地球科学和许多其他领域。 在这个项目中，正在开发的求解偏微分方程的方法将有助于解决涉及波传播，流体流动建模，甚至层析成像的问题。 球面基函数和紧球面框架的工作将影响地球科学，特别是在拟合通过卫星或地面站在地球上收集的大型分散数据集的情况下。

The overall goal of this project is to develop new methods and tools for the analysis and synthesis of scattered data by means of radial basis functions, spherical basis functions, and tight spherical frames. The research will provide a foundation for the effectiveness of various radial basis function meshless methods for solving partial differential equations by estimating the errors made in using them; knowing these errors can then serve as a guide in creating and implementing algorithms based on radial basis and spherical basis functions, both for meshless methods for solving differential equations and for scattered data surface fitting. Tight, well-localized spherical frames are a newly discovered tool for dealing with scattered data on spheres, and are similar to frames used in connection with wavelet analysis. Very little is known about the specifics of these tight frames, and a second goal of this project is to develop them, and algorithms based on them, to efficiently process scattered data on spheres.Problems involving analyzing and synthesizing data taken from scattered sites arise in diverse fields -- computer-aided design graphics, data mining, medical imaging, learning networks, geoscience, and many other areas. The methods under development in this project for solving partial differential equations will help in solving problems involving wave propagation, modeling fluid flow, and even tomography. The work on spherical basis functions and tight spherical frames will impact geoscience, especially in the case of fitting large scattered-data sets collected over the earth via satellites or by ground stations.