Topics in Approximation Theory

逼近论专题

• 批准号: 9803501
• 负责人: Marian Neamtu
• 金额: $ 6.64万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803501&HistoricalAwards=false
• 关键词: Topics; Approximation; Theory

This research project addresses a number of problems in applications of approximation theory and spline functions. The problems are grouped into four broad areas: multivariate approximation, subdivision algorithms and refinement equations, shape preserving interpolation and approximation, and harmonic finite elements for solving partial differential equations in spherical geometry. Understanding of approximation properties of the spline spaces to be considered in this research is of fundamental importance in various applications, such as the finite element method. While the primary interest will be in spaces defined on non-uniform partitions (such as spaces of piecewise polynomials on planar and spherical triangulations), a large portion of the research will also be devoted to the important special case of shift-invariant spaces. Some of the obtained theoretical results will be applied in the construction of finite elements suitable for solving partial differential equations whose domain is the sphere. Examples are the shallow water equations, nondivergent barotropic vorticity equation, and altimetry-gravimetry equations of physical geodesy. Accurate solutions of those equations are of vital importance for modeling and understanding physical phenomena such as the motion of the atmosphere, evolution of the Earth's mantle and outer core, and the dynamics of the Earth's magnetic field.

这项研究项目解决了逼近理论和样条函数应用中的一些问题。这些问题可分为四大类：多元逼近、细分算法和精化方程、保形插补和逼近，以及球面几何中偏微分方程组的调和有限元。理解本研究中要考虑的样条空间的逼近性质在各种应用中是非常重要的，例如有限元方法。虽然主要的兴趣将是定义在非均匀划分上的空间(例如平面三角剖分和球面三角剖分上的分段多项式空间)，但很大一部分研究也将致力于移位不变空间的重要特例。所得到的一些理论结果将被应用于构造适合于求解区域为球面的偏微分方程组的有限元。例如物理大地测量中的浅水方程、无辐散正压涡度方程和测高-重力方程。这些方程的精确解对于模拟和理解物理现象至关重要，例如大气运动、地幔和地核的演化以及地球磁场的动力学。