Approximations and Modeling with Zonal Functions on Spheres and Euclidean Spaces

Ragozin教授计划研究与多元多项式和广义样条（能量最小化）近似，数值积分以及多变量势和相关函数的快速计算方法相关的几个主题。这些研究将旨在开发近似（大型）有限数据集之间的函数关系的方法，当底层数据点从有限维球体或其他具有高度齐次性的紧致空间或从高维欧几里德空间绘制时。球体和欧几里得空间的高度对称性和同质性，以及径向函数的使用，在很大程度上尊重对称性和同质性，为不同的研究提供了一个主要的统一特征。在这些领域中，存在一组定义良好且可有效计算的（不变）多项式核，它们依赖于比底层空间少得多的参数，但可以并且将用于开发追求的近似，积分和快速计算过程。球面和欧几里得空间上显式逼近过程的发展，以及这些过程中误差的可计算边界，将增强实际科学计算的基础设施。依靠径向函数展开，基于分散数据集的实际计算变得可行。欧几里德空间中多谐径向函数的快速计算方法将扩展现有的薄板样条在普通三维空间中的方法，并促进多元拉普拉斯样条插值和平滑技术的有效应用，包括交叉验证，以极大的数据集。许多科学和工程问题涉及对两个、三个或更多连续量之间的复杂函数关系进行建模。这项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，并估计这些近似与事实的接近程度。通过使用简单函数的组合，这些简单函数只依赖于空间中两点之间的距离或球体（如地球）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，现有的基于精确建模的方法需要对包含100,000个点的集合进行千万亿次操作。基于这里开发的近似值的方法将减少十亿倍，因此只需要几百万个操作。今天的计算机可以迅速处理这么多的操作。

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.