Analysis and Synthesis of Scattered Data on Surfaces via Radial and Related Basis Functions

本课题的主要目标是开发利用径向和相关基函数对散射数据进行分析和综合的新方法和工具。最近，研究人员Ward和Narcowich及其合作者引入了一种新工具，用于将3D中包含的2d可定向表面的无散度矢量场切线拟合到表面上分散位置的此类场的样本中。该方法包含一个由径向基函数（rbf）构造的核，可用于解决地球物理学中的问题，例如，单个静水大气层中不可压缩流体的非线性流动。数学上，不可压缩性转化为表面散度消失的速度。在这些情况下，将无发散的切矢量场拟合到数据中，将有助于对所涉及的不可压缩速度场进行建模。除了处理分散的数据外，它还具有避免对极点进行特殊处理的优点。研究人员研究了该工具的近似特性，以估计其数据拟合误差以及它的鲁棒性。在某些情况下，数值实验表明数据的拟合和鲁棒性都很好。此外，非结构化的分散数据——即聚集或聚类或非均匀分布的数据——对任何方法都存在问题，包括基于一个条件正定函数的翻译的传统rbf类型算法。这些聚类数据自然地出现在学习理论、神经网络、无网格方法和其他情况中。为了处理聚类数据，研究人员开始研究新基、前置条件的构建，以及一种新的rbf类型范式。涉及分析从空间或地球表面的分散地点（即不规则地点）获取的数据的问题经常出现在不同的领域：计算机辅助设计图形、数据挖掘、医学成像、学习网络和地球科学，以及许多其他领域。这些问题给传统的方法带来了困难，传统的方法是基于在统一的地点收集数据。例如，天气预报是基于一个数学模型，假设地球表面是一个球体的表面。研究人员开发了新的方法和工具，通过所谓的径向基函数方法来帮助分析这种现象。

WardDMS-0807033 The main object of this project is to develop new methodsand tools for the analysis and synthesis of scattered data bymeans of radial and related basis functions. Very recently, theinvestigators Ward and Narcowich and a collaborator introduced anew tool for fitting a divergence-free vector field tangent to a2D orientable surface contained in 3D to samples of such a fieldtaken at scattered sites on the surface. The method, whichinvolves a kernel constructed from radial basis functions (RBFs),has applications to problems in geophysics -- for instance, thenonlinear flow of an incompressible fluid in a single hydrostaticatmospheric layer. Mathematically, incompressibility translatesto the velocity having vanishing surface divergence. Fittingdivergence-free tangent vector fields to data taken in thesecases would help in modeling the incompressible velocity fieldsinvolved. In addition to handling scattered data, it has theadvantage of avoiding special treatment for the poles. Theinvestigators study the approximation properties of this tool,with a view towards estimates on its data-fitting errors as wellhow robust it is. For certain cases, numerical experimentssuggest both a good fit of data and robustness. In addition,unstructured scattered data -- i.e., clumped, or clustered, ornonuniformly distributed data -- present problems for any method,including the traditional RBF-type algorithms based on translatesof one conditionally positive definite function. These clustereddata occur naturally in learning theory, neural nets, meshlessmethods, and other situations. To deal with clustered data, theinvestigators initiate the study of the construction of newbases, preconditioners, and in general a new RBF-type paradigm. Problems that involve analyzing data taken from scatteredsites -- that is, irregularly placed sites -- in space or on thesurface of the earth arise frequently in diverse fields:computer-aided design graphics, data mining, medical imaging,learning networks, and geoscience, in addition to many otherareas. Such problems present difficulties for traditionalmethods, which are based on collecting data at uniformly placedsites. For example, weather prediction is based on amathematical model where the earth's surface is assumed to be asurface of a sphere. The investigators develop new methods andtools to help in analyzing such phenomena via so-called radialbasis function methods.