Cubature rules and Approximation on Regular Domains

正则域上的体积规则和近似

• 批准号: 1106113
• 负责人: Yuan Xu
• 金额: $ 14.95万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106113&HistoricalAwards=false
• 关键词: Cubature; rules; Approximation; Regular; Domains

XuDMS-1106113 The principal investigator studies cubature rules, which are numerical integration formulas for higher dimensional integrals, and approximation of functions on regular domains such as cubes, balls, spheres and simplexes. The project combines several topics: numerical analysis, discrete Fourier analysis, orthogonal polynomials, and approximation theory. Two approaches are emphasized. The first approach is based on a connection between cubature rules and discrete Fourier analysis with translation tiling. The approach allows one to study, in several stages, cubature rules and interpolation by exponential functions on the fundamental domain of the translation tiling, by trigonometric functions on the fundamental simplex of the domain, and by algebraic polynomials on corresponding domains, and it yields results on cubature rules, interpolation, orthogonal polynomials and approximation. The second approach starts with a characterization of best approximation by polynomials on the sphere and on the ball in terms of the smoothness of the functions being approximated, while the smoothness is measured by the differences of the function values in Euler angles. This line of work is based on recent results of the investigator and his collaborators that for some problems it is necessary to work with these angles, even though their number is much larger than the dimension. Cubature rules, which are multidimensional numerical integration formulas, and approximation on regular domains in higher dimensional spaces are fundamental tools in a variety of applications, because most integrals can only be evaluated numerically and very few problems can be evaluated exactly. At the current stage, in contrast to the situation in one dimension, many fundamental problems in these two areas have not been resolved, despite increasing need for them in applications. The project aims at finding new methods to construct numerically efficient algorithms, such as accurate cubature rules with fewer nodes, fast discrete Fourier transforms, and approximation operators on regular domains such as cubes, balls, spheres and simplexes. The algorithms have applications in scientific computing, imaging, statistics, and geosciences.

徐DMS-1106113 主要研究人员研究cubature规则，这是高维积分的数值积分公式，以及在立方体，球，球体和单形等规则域上的函数近似。 该项目结合了几个主题：数值分析，离散傅立叶分析，正交多项式和逼近理论。 强调两种方法。 第一种方法是基于体积规则和离散傅立叶分析与平移平铺之间的连接。 该方法允许一个研究，在几个阶段，容积规则和插值指数函数的基本域的平移平铺，三角函数的基本单形的域，并通过代数多项式相应的域，它产生的结果容积规则，插值，正交多项式和近似。 第二种方法开始与最佳逼近的表征多项式的领域和球上的平滑度的函数被近似，而平滑度是由欧拉角的函数值的差异来衡量。 这一系列的工作是基于最近的研究结果和他的合作者，对于某些问题，有必要与这些角度，即使他们的数量是远远大于尺寸。 立方规则，这是多维数值积分公式，以及在高维空间中的正则域上的近似是各种应用中的基本工具，因为大多数积分只能数值计算，很少有问题可以精确计算。 在现阶段，与一维的情况相反，这两个领域的许多基本问题尚未解决，尽管在应用中对它们的需求越来越大。 该项目旨在寻找新的方法来构建数值高效算法，例如具有更少节点的精确体积规则，快速离散傅立叶变换以及立方体，球，球体和单形等规则域上的近似运算符。 这些算法在科学计算、成像、统计和地球科学中有应用。