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

基本信息

  • 批准号:
    RGPIN-2020-03909
  • 负责人:
  • 金额:
    $ 1.97万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2020
  • 资助国家:
    加拿大
  • 起止时间:
    2020-01-01 至 2021-12-31
  • 项目状态:
    已结题

项目摘要

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.
我的研究的主要目标是调查抽样离散化,容积公式和它们的连接与定量近似理论在高维设置。 采样离散化是指通过固定的有限点集上的函数值将连续对象转换为离散对象的过程,而容积法是指通过 函数值在有限点集上的加权和,称为求积公式。离散化是使连续问题在计算上可行的重要步骤,而体积公式在离散化和计算中起着至关重要的作用。 高维积分的实用计算。 为了确保问题可以被计算机访问,离散化的第一步通常是通过低维对应物(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)从高维子空间采样离散函数的Lq范数;(ii)高维函数空间中的偏差估计和体积公式;(iii)能量最小化和最优体积公式之间的相互作用; (iv)之间的连接 球面及相关区域上的局部支撑正定函数; (v)良好分布的点集(低差异,cubature,能量最小化,晶格)的新结构;(六)高维域近似的无量纲估计。 我的研究 将使用一种新技术, 强大的概率技术,基于链式和大偏差不等式, 导致多维近似理论(例如,估计 熵数和N宽度,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。

项目成果

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Dai, Feng其他文献

Perineural Methylprednisolone Depot Formulation Decreases Opioid Consumption After Total Knee Arthroplasty.
全膝关节置换术后,周围甲基丙诺酮仓库配方可降低阿片类药物的消耗。
  • DOI:
    10.2147/jpr.s378243
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    2.7
  • 作者:
    Del Toro-Pagan, Nicole M.;Dai, Feng;Banack, Trevor;Berlin, Jill;Makadia, Satya A.;Rubin, Lee E.;Zhou, Bin;Huynh, Phu;Li, Jinlei
  • 通讯作者:
    Li, Jinlei
Dynamic Response and Failure Mechanism of Brittle Rocks Under Combined Compression-Shear Loading Experiments
脆性岩石压剪联合加载试验动力响应及破坏机制
Numerical investigation on the dynamic progressive fracture mechanism of cracked chevron notched semi-circular bend specimens in split Hopkinson pressure bar tests
霍普金森压杆试验中人字形缺口半圆形弯曲试件动态渐进断裂机制的数值研究
  • DOI:
    10.1016/j.engfracmech.2017.09.001
  • 发表时间:
    2017-10-15
  • 期刊:
  • 影响因子:
    5.4
  • 作者:
    Du, Hongbo;Dai, Feng;Xu, Yuan
  • 通讯作者:
    Xu, Yuan
Some Fundamental Issues in Dynamic Compression and Tension Tests of Rocks Using Split Hopkinson Pressure Bar
  • DOI:
    10.1007/s00603-010-0091-8
  • 发表时间:
    2010-11-01
  • 期刊:
  • 影响因子:
    6.2
  • 作者:
    Dai, Feng;Huang, Sheng;Tan, Zhuoying
  • 通讯作者:
    Tan, Zhuoying
Association of low-level lead exposure with all-cause and cardiovascular disease mortality in US adults with hypertension: evidence from the National Health and Nutrition Examination Survey 2003-2010.
  • DOI:
    10.1186/s13690-023-01148-6
  • 发表时间:
    2023-08-14
  • 期刊:
  • 影响因子:
    3.3
  • 作者:
    Wang, Lili;Wang, Chaofan;Liu, Tao;Xuan, Haochen;Li, Xiaoqun;Shi, Xiangxiang;Dai, Feng;Chen, Junhong;Li, Dongye;Xu, Tongda
  • 通讯作者:
    Xu, Tongda

Dai, Feng的其他文献

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{{ truncateString('Dai, Feng', 18)}}的其他基金

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
  • 财政年份:
    2022
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
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
  • 财政年份:
    2021
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains
正则域上的体积公式、正交展开和定量逼近
  • 批准号:
    RGPIN-2015-04702
  • 财政年份:
    2019
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains
正则域上的体积公式、正交展开和定量逼近
  • 批准号:
    RGPIN-2015-04702
  • 财政年份:
    2018
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains
正则域上的体积公式、正交展开和定量逼近
  • 批准号:
    RGPIN-2015-04702
  • 财政年份:
    2017
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains
正则域上的体积公式、正交展开和定量逼近
  • 批准号:
    RGPIN-2015-04702
  • 财政年份:
    2016
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains
正则域上的体积公式、正交展开和定量逼近
  • 批准号:
    RGPIN-2015-04702
  • 财政年份:
    2015
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Orthogonal expansions, cubature formulas and approximation in several variables
正交展开、体积公式和多变量近似
  • 批准号:
    311678-2010
  • 财政年份:
    2014
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Orthogonal expansions, cubature formulas and approximation in several variables
正交展开、体积公式和多变量近似
  • 批准号:
    311678-2010
  • 财政年份:
    2013
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual
Orthogonal expansions, cubature formulas and approximation in several variables
正交展开、体积公式和多变量近似
  • 批准号:
    311678-2010
  • 财政年份:
    2012
  • 资助金额:
    $ 1.97万
  • 项目类别:
    Discovery Grants Program - Individual

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