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

WARD DMS-0807033这个项目的主要目标是开发新的方法和工具，用于利用径向和相关的基函数来分析和综合散乱的数据。最近，研究人员Ward和Narcowich及其合作者引入了一种新的工具，用于将与3D中包含的2D可定向曲面相切的无散度矢量场与在表面上分散位置拍摄的这种场的样本进行拟合。该方法涉及由径向基函数(RBF)构造的核，可应用于地球物理中的问题--例如，不可压缩流体在单个流体静力大气层中的非线性流动。从数学上讲，不可压缩性转化为表面散度为零的速度。将无散度正切向量场拟合到这些情况下的数据将有助于对所涉及的不可压缩速度场进行建模。除了处理零散的数据外，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性质，以期对其数据拟合误差以及它的健壮性进行估计。在某些情况下，数值实验表明数据具有很好的拟合能力，而且具有很好的稳健性。此外，非结构化的散乱数据--即聚集、聚集或非均匀分布的数据--给任何方法带来了问题，包括传统的基于一个条件正定函数的翻译的RBF型算法。这些聚集的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下。为了处理聚集的数据，研究人员开始了对新碱基、预条件库和一般新的RBF类型范式的构建的研究。涉及分析从空间或地球表面的分散地点--即不规则布置的地点--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学、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，这些方法是基于在统一的地点收集数据的。例如，天气预报是基于一个数学模型，该模型假定地球表面是球体的表面。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.