Cubature rules and Approximation on Regular Domains

XUDMS-1106113主要研究立方体规则，这是高维积分的数值积分公式，以及正则区域上函数的逼近，如立方体、球、球和单纯形。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。重点介绍了两种方法。第一种方法是基于体积规则和带有平移平铺的离散傅立叶分析之间的联系。这种方法允许分阶段地研究平移平铺的基本区域上的指数函数的体积规则和插值，区域的基本单纯形上的三角函数和相应区域上的代数多项式，并得到关于体积规则、内插、正交多项式和逼近的结果。第二种方法首先用被逼近函数的光滑度来刻画球面上和球面上多项式的最佳逼近，而光滑度是通过函数在欧拉角上的差值来衡量的。这一工作是基于研究人员和他的合作者最近的结果，即对于某些问题，有必要处理这些角度，即使它们的数量远远大于维度。体积规则是高维空间中的多维数值积分公式，在高维空间中的正则区域上的逼近是各种应用中的基本工具，因为大多数积分只能用数值计算，很少有问题可以精确计算。在现阶段，与一个层面的情况相比，这两个领域的许多根本问题尚未得到解决，尽管在应用中越来越需要它们。该项目旨在寻找新的方法来构造数值高效的算法，例如节点较少的精确立方规则、快速离散傅立叶变换以及规则区域上的近似算子，如立方体、球、球体和单纯形。这些算法在科学计算、成像、统计学和地学中都有应用。

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.