Mathematical Sciences: Multivariate Spline Approximation & Multivariate Polynomial Interpolation

在这个项目中，首席研究员将在数值分析和逼近理论中追求两个相辅相成的领域，即多元样条逼近和多元多项式插值。关于第一个领域，首席研究员将继续他在规则网格和非规则网格上的多元样条的研究。他还将研究样条与径向基函数和小波之间的联系。关于第二个领域，首席研究员将进一步扩展他最近发现的一种简单而通用的方法，即对于给定的S变量中的有限点集，选择一个在这些点处进行插值的“好”多项式空间。微分方程组和逼近理论中的许多问题都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在理论工作的基础上发展出来的，这些基函数可以用来表示一组离散点上的函数。基函数在非常真实的意义上是“积木”，连续函数就是从这些积木中创建出来的。在这个项目中，首席研究员将研究被称为“样条基”的基函数，这种基函数在解决许多应用领域的问题时非常有用。

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.