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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739555
• 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.

本文的主要研究目标是探讨高维情形下的抽样离散化、容积公式及其与定量逼近理论的联系。抽样离散化是指在一个固定的有限点集上通过函数值将连续对象转化为离散对象的过程，而体积法是指通过有限点集上的函数值的加权和来数值地近似函数的多维积分的方法，这就是所谓的容积公式。离散化是使连续问题在计算上可行的重要步骤，而容积公式在高维积分的离散化和实际计算中起着至关重要的作用。为了确保问题可以被计算机访问，离散化的第一步通常是用低维对应物（例如多元多项式，样条，神经网络）近似连续算子和高维函数空间的过程。在这个程序中，我将集中在近似和离散化过程中不可避免地出现的误差的定量估计。 许多问题的最重要的离散化，容积公式和近似理论在更高的维度仍然是开放的。该计划所研究的问题包括：（i）高维子空间中函数Lq范数的抽样离散化;（ii）高维函数空间中的偏差估计和容积公式;（iii）能量最小化和最优容积公式之间的相互作用;（iv）球面上局部支集正定函数与相关区域之间的联系;（iv）球面上局部支集正定函数与相关区域之间的联系;（v）球面上局部支集正定函数与相关区域之间的联系;（v）球面上局部支集正定函数与相关区域之间的联系;（v）球面上局部支集正定函数与相关区域之间的联系。（v）良好分布点集的新构造（低差异、体积、能量最小化、格）;（vi）高维域上近似的无维估计。 我的研究将使用一种新技术，该技术结合了基于链式和大偏差不等式的强大概率技术，以及多维近似理论中的各种深入结果（例如，熵数和N宽度的估计，各种多项式不等式，多项式近似中的直接和逆无量纲估计）。我也期望经典的正交多项式展开式理论，特别是球调和分析，将成为我研究的有力工具。 该程序与计算数学（即，数值积分方法）、概率论、统计学、人工智能和其他数学领域。该计划的科学成果预计将影响数学的几个领域，丰富和交叉它们的新成果，思想和方法。