Sampling discretization, cubature formulas and quantitative approximation in multidimensional settings

多维环境中的采样离散化、体积公式和定量近似

• 批准号: RGPIN-2020-03909
• 负责人: Dai, Feng
• 金额: $ 1.97万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758337
• 关键词: Sampling; discretization; cubature; formulas; quantitative

The primary goal of my research is to investigate sampling discretization, cubature formulas  and their connections with quantitative approximation theory in higher dimensional settings.  Sampling discretization refers to the process of transferring continuous objects into their discrete counterparts through function values at a fixed finite set of points, whereas cubature refers to a method for numerically approximating a multidimensional integral of a function through a  weighted sum of function values on a finite set of points, which is called a cubature formula. Discretization is an important step in making a continuous problem computationally feasible, while cubature formulas have been playing crucial roles in discretization and practical evaluation of high dimensional integrals. To ensure the problems are accessible by computers, the first step towards discretization is usually the process of approximating continuous operators and higher dimensional function spaces by lower dimensional counterparts (e.g. multivariate polynomials, splines, neural networks). In this program, I will focus on quantitative estimates of errors that inevitably arise in the process of approximation and discretization.    Many questions of utmost importance to discretization, cubature formulas and approximation theory in higher dimensions remain wide open. The questions under investigation in this program include (i)  sampling discretizations of Lq norms of functions from a high-dimensional subspace; (ii) discrepancy estimates and cubature formulas in high-dimensional function spaces; (iii) interplay between energy minimization and optimal cubature formulas; (iv) connections between locally supported positive definite functions on spheres and related domains; (v) new constructions of well-distributed point sets (low-discrepancy, cubature, energy-minimizing, lattices); and (vi) dimension-free estimates for approximation on high-dimensional domains.     My research will use a new technique, which combines powerful probabilistic techniques, based on chaining and large deviation inequalities, with various deep results in multidimensional approximation theory (e.g., estimates of entropy numbers and N-widths, various polynomial inequalities,  direct and inverse dimension-free  estimates in polynomial approximation). I also expect that the theory of classical orthogonal polynomial expansions, especially spherical harmonic analysis,  will be a powerful  tool in my research.    This program is connected to computational mathematics (i.e., the methods of numerical integration), probability, statistics, artificial intelligence and other areas of mathematics. The scientific outputs of this program are expected to impact several areas of mathematics, enriching and cross-fertilizing them with new results, ideas, and methods.

我的研究的主要目的是研究采样离散化，立方体公式及其与较高维度环境中定量近似理论的连接。采样离散化是指通过固定有限点的函数值将连续对象传输到离散对应物中的过程，而立方体是指通过数值近似于通过在有限点上的加权函数值来近似函数的多维数字积分的方法，这是一个有限的点，称为cubature Formula。离散化是在计算上可行的连续问题的重要一步，而立方公式在离散化和对高维积分的实际评估中发挥了关键作用。为了确保计算机可以解决问题，离散化的第一步通常是通过较低维度对应物（例如，多元多项式，细条，神经网络）近似连续运算符和较高维度函数空间的过程。在此程序中，我将重点介绍在近似和离散过程中不可避免地出现的错误的定量估计。对于离散化，较高维度中的立方公式和近似理论的许多最重要的问题仍然是敞开的。该计划中所研究的问题包括（i）从高维子空间中对功能的LQ规范进行抽样； （ii）高维函数空间中的差异估计和立方公式； （iii）能量最小化和最佳立方体公式之间的相互作用； （iv）在球体和相关域上局部支持的正定义函数之间的连接； （v）分布良好的点集的新结构（低静止，立方体，能量最小化，晶格）； （vi）高维域上近似值的无维估计。我的研究将使用一种新技术，该技术基于链式和较大的偏差不平等，结合了强大的概率技术，并在多维近似理论（例如，熵数量和N宽度的估计值，各种多项式不平等，直接和逆尺寸估计值估计）中的各种深层结果。我还期望经典的正交多项式扩展，尤其是球形谐波分析的理论将是我研究中的一个强大工具。该程序连接到计算数学（即数值集成方法），概率，统计，人工智能和其他数学领域。预计该计划的科学输出将影响数学的几个领域，并通过新的结果，想法和方法丰富和交叉侵占。