Mathematical Sciences: Miltivariate Interpolation and Approximation

数学科学：多元插值和逼近

• 批准号: 9203859
• 负责人: Peter Alfeld
• 金额: $ 1.2万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1995-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203859&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Miltivariate; Interpolation; Approximation

Combining techniques from disparate areas (Algebra, Geometry,and Combinatorics) the investigator and his coworkers recently determined the generic dimension of trivariate spline spaces (of sufficiently high degree) defined on tetrahedral decompositions. In this project the techniques will be further developed and applied to multivariate spline research, particularly as regards the following problems: exact dimension formulas, improved upper and lower bounds on dimensions, dependence of the dimension on geometric degeneracies, existence and construction of local bases,identification of useful sub- and superspaces of the full spline space, solvability of interpolation problems, effects of degeneracies on numerical stability and solvability of interpolation problems, algorithms, surface design, approximation of functions defined on surfaces, and investigation of domain partitions other than simplicial ones. Splines are smooth piecewise polynomial functions. They are used ubiquitously to solve problems involving functions of one independent variable. Regarding functions of several independent variables, splines (which in this context are called "multivariate splines") are much less understood and their structure is vastly more complicated. Yet they have great potential for applications such as the solution of partial differential equations (e.g., modeling fluid flow, heat distribution, combustion, or radioactive decay), interpolation and approximation of data, and the design of shapes (e.g., of a vehicle). They are also fundamental objects that deserve to be studied in their own right.

结合不同领域（代数、几何和组合学）的技术，研究者和他的同事最近确定了在四面体分解上定义的三变量样条空间（足够高的程度）的通用维数。 在这个项目中，这些技术将进一步发展并应用于多元样条研究，特别是关于以下问题：精确的尺寸公式、改进的尺寸上限和下限、尺寸对几何简并性的依赖性、局部基的存在和构造、完整样条空间的有用子空间和超空间的识别、插值问题的可解性、简并性对插值问题的数值稳定性和可解性的影响、算法、曲面设计、定义的函数的近似曲面，以及除单纯域划分之外的域划分的研究。 样条曲线是平滑的分段多项式函数。 它们普遍用于解决涉及一个自变量函数的问题。 对于几个自变量的函数，样条曲线（在本文中称为“多元样条曲线”）了解得很少，而且它们的结构要复杂得多。 然而，它们在诸如偏微分方程的求解（例如流体流动、热分布、燃烧或放射性衰变建模）、数据插值和近似以及形状设计（例如车辆）等方面具有巨大的应用潜力。 。 它们也是值得研究的基本对象。