Approximations and Modeling with Zonal Functions on Spheres and Euclidean Spaces

球体和欧几里得空间上的分区函数的近似和建模

• 批准号: 9972004
• 负责人: David Ragozin
• 金额: $ 5.2万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9972004&HistoricalAwards=false
• 关键词: Approximations; Modeling; Zonal; Functions; Spheres

Professor Ragozin plans to investigate several topics connected withmultivariate polynomial and generalized spline (energy minimizing)approximations, numerical integration, and fast computational methods for multivariable potentials and related functions. These investigationswill aim to develop methods for approximating functional relationshipsamong (large) finite data sets when the underlying data points are drawn from finite dimensional spheres or other compact spaces with a high degree of homogeneity, or from high dimensional Euclidean spaces. The high degree of symmetry and homogeneity of spheres and Euclidean spaces,and the use of radial functions, which respect much of the symmetry and homogeneity, provide a major unifying feature of the different investigations. In each of these domains there exist a well defined and effectively computable set of (invariant) polynomial kernels whichdepend on many fewer parameters then the underlying space, but which can and will be used to develop the sought after approximation, integration and fast computational processes. The development of explicit approximation processes on spheres and Euclidean spaces, together with computable bounds on the errors in these processes, will enhance the infrastructure for practical scientific computation. By reliance on radial function expansions, practical computations based on scattered data sets become feasible. The fast computational methods forpolyharmonic radial functions in Euclidean spaces will extend existingmethods for thin-plate splines in ordinary three space, and facilitateeffective application of multivariate Laplacian spline interpolation andsmoothing techniques, including cross-validation, to extremely large data sets.Many scientific and engineering problems involve modelling the complexfunctional relationships between two, three or more continuousquantities. The aim of this research is to develop new mathematicaltechniques to approximate such relationships by simpler functions,together with estimates for how close these approximations come to thetruth. By use of combinations of simple functions which depend only onthe distance between two points in space or on the surface of a spheresuch as the earth, this work hopes to provide highly efficient means ofapproximation and modelling; computational codes which exploit thesemethods should be able to deal in practical time with extremely largedata sets. For example, existing methods based on exact modelling require several quadrillion operations for sets containing 100,000 points. Methods based on the approximations developed here will reduce this by afactor of a billion, so only several million operations are required.Todays computers can rapidly handle this number of operations.

Ragozin教授计划研究多变量多项式和广义样条（能量最小化）近似，数值积分和多变量势和相关函数的快速计算方法。 这些investigationwill的目的是开发方法近似的功能relationshipsamong（大）有限的数据集时，基本的数据点是从有限维领域或其他紧凑的空间具有高度的同质性，或从高维欧氏空间。 高度的对称性和均匀性的领域和欧几里德空间，并使用径向函数，其中尊重大部分的对称性和均匀性，提供了一个主要的统一功能的不同调查。在这些域中的每一个存在一个定义良好的和有效的可计算的（不变的）多项式内核，它依赖于少得多的参数，然后潜在的空间，但它可以并将被用来开发追求后的近似，积分和快速计算过程。球和欧氏空间上的显式近似过程的发展，以及这些过程中误差的可计算范围，将增强实际科学计算的基础设施。 通过依赖径向函数展开，基于分散数据集的实际计算变得可行。 欧氏空间中多调和径向函数的快速计算方法将扩展现有的三次空间中薄板样条的计算方法，并促进多元Laplacian样条插值和光滑技术（包括交叉验证）在超大数据集上的有效应用。本研究的目的是开发新的数学技术，通过更简单的函数来近似这种关系，以及估计这些近似值与真实值的接近程度。通过使用简单函数的组合，这些函数只依赖于空间中两点之间的距离或地球等球体表面上的距离，这项工作希望提供高效的近似和建模方法;利用这些方法的计算代码应该能够在实际时间内处理非常大的数据集。 例如，基于精确建模的现有方法需要对包含100，000个点的集合进行几次四边形运算。基于这里提出的近似方法将把这一过程减少十亿倍，因此只需要几百万次运算，今天的计算机可以迅速处理这么多的运算。