Mathematical Sciences: Multivariate Spline Approximation & Multivariate Polynomial Interpolation

在这个项目中，首席研究员将追踪两个 数值分析与近似的互补领域 理论，即多元样条逼近和 多元多项式插值 关于第一 在这一领域，首席研究员将继续研究 规则和不规则网格上的多元样条。 他将 还研究了样条函数和径向基函数之间的关系 函数和小波 关于第二个领域， 首席研究员将进一步扩大他最近的发现 一个简单而通用的选择方法，对于给定的有限 s变量中的一组点，一个“好”的多项式空间， 在这些点上进行插值。 微分方程与逼近中的许多问题 理论需要使用数值算法来构造 函数的精确表示。 这些算法中的许多 都是从理论工作中发展出来的， 函数，可用于表示一组 离散点 基函数在非常真实的意义上是 “积木”，从其中创建连续的功能。 在本项目中，主要研究者将研究基础 函数称为“样条”，这是非常有用的解决 一些应用领域的问题。

In this project the principal investigator will pursue two complementary areas in numerical analysis and approximation theory, namely, multivariate spline approximation and multivariate polynomial interpolation. With regard to the first area, the principal investigator will continue his study of multivariate splines on regular and irregular meshes. He will also look into the connections between splines and radial basis functions and wavelets. With regard to the second area, the principal investigator will further expand his recent discovery of a simple and general method of choosing, for a given finite set of points in s variables, a "good" polynomial space for interpolation at those points. Many problems in differential equations and approximation theory require the use of numerical algorithms to construct accurate representations of functions. Many of these algorithms have developed out of theoretical work on the kinds of basis functions that can be used to represent functions at a set of discrete points. Basis functions are in a very real sense the "building blocks" out of which continuous functions are created. In this project the principal investigator will study basis functions called "splines" that are very useful for solving problems in a number of applied areas.