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

WardDMS-0807033 本项目的主要目标是开发新的方法和工具，通过径向基函数和相关基函数分析和综合散乱数据。 最近，研究人员沃德和Narcowich和一个合作者介绍了一种新的工具，用于拟合一个无发散向量场的切线到一个2D定向表面包含在3D的样本，这样一个字段采取在表面上分散的网站。 该方法，它涉及到一个核心构造的径向基函数（RBFs），已应用于物理学的问题-例如，非线性流动的不可压缩流体在一个单一的hydrostaticatmospheric层。 在数学上，不可压缩性等于表面发散消失的速度。 拟合发散自由切向量场的数据，在这些情况下，将有助于模拟不可压缩的速度场。 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，以期估计其数据拟合误差以及它的鲁棒性。 对于某些情况下，数值实验表明，这两个良好的数据拟合和鲁棒性。 此外，非结构化的分散数据--即，聚集的，或群集的，或非均匀分布的数据--对任何方法都存在问题，包括传统的基于一个条件正定函数的RBF型算法。 这些聚类数据自然出现在学习理论，神经网络，无网格方法和其他情况下。 为了处理聚类数据，研究者们开始研究新的基、预条件子和一个新的RBF型范式的构建。 涉及分析从空间或地球表面的分散站点（即不规则放置的站点）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均匀放置的地点收集数据。 例如，天气预报是基于数学模型，其中地球的表面被假定为一个球面。 研究人员开发了新的方法和工具，以帮助通过所谓的径向基函数方法分析这种现象。

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.