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Nonlinear Approximation in Geometric, Harmonic, and Anisotropic Settings with Applications

几何、谐波和各向异性设置中的非线性近似及其应用

基本信息

批准号：
1714369
负责人：
Pencho Petrushev
金额：
$ 16.29万
依托单位：
University of South Carolina at Columbia
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-15 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714369&HistoricalAwards=false
关键词：
Nonlinear Approximation Geometric Harmonic Anisotropic

项目摘要

1714369Petrushev Many areas ranging from physics, geodesy and geomagnetism to cosmology and to data analysis require efficient representation and approximation of the underlying functions in the natural topology of the targeted application. The capturing of physical phenomena and data structure occurring at various scales requires approximation from locally supported multiscale systems relative to the application domains. Moreover, these approximation methods should be amenable to fast and accurate computation. Thus project aims at increasing our fundamental understanding of nonlinear approximation theory and its applications in three main directions. The first objective is to develop nonlinear approximation theory in various geometric and nonclassical settings from multiscale systems that are well adapted to the targeted applications. The second aim is to study the approximation of harmonic functions from shifts of the Newtonian potential, with targeted applications to geodesy, geomagnetism, and physics. The third goal is to approximate functions that are smooth on domains in space separated by smooth curves or surfaces. Here the idea is to use adaptively anisotropic multiscale dilations of the space, which enable the approximation tool to adjust to curved singularities. A core objective of this project is the development of nonlinear n-term approximation from frames and other systems in various geometric and nonclassical settings, such as on the sphere, ball, box, and simplex with weights, as well as in the context of Lie groups and Riemannian manifolds. All these settings are covered by the general framework of Dirichlet spaces with heat kernel having Gaussian bounds. The key point of the approach is to give us the freedom of dealing with (a) different geometries, (b) compact and noncompact spaces, and (c) spaces with nontrivial weights, and at the same time to allow for the development and frame decomposition of Besov and Triebel-Lizorkin spaces with complete range of indices. The development of the underlying heat kernel theory and nonlinear n-term approximation from localized systems are basic aspects of this theory. Another goal of this project is the development of nonlinear approximation of harmonic functions on the d-dimensional ball from linear combinations of shifts of the Newtonian potential. This includes the complete characterization of the rates of approximation and the development of an effective algorithm that achieves the rates of best approximation. Anisotropic phenomena appear in various contexts in analysis, partial differential equations, and in applications. For instance, functions are frequently very smooth on domains in space separated by smooth curves or manifolds. The project aims at resolving this kind of singularity of functions by using the framework of anisotropic multiscale dilations, which may change rapidly from point to point at any level and in depth. The main strands of this approach are (i) the development of an algorithm for rapid construction of best or near best dilation matrices leading to optimal sparsity, (ii) the construction of highly localized anisotropic frames and their use in nonlinear approximation of functions.

小行星1714369 从物理学、大地测量学和地磁学到宇宙学和数据分析的许多领域都需要在目标应用的自然拓扑结构中有效地表示和近似底层函数。捕获发生在不同尺度的物理现象和数据结构，需要从本地支持的多尺度系统相对于应用领域的近似。此外，这些近似方法应该能够进行快速和准确的计算。因此，该项目旨在增加我们对非线性逼近理论及其在三个主要方向上的应用的基本理解。第一个目标是发展非线性逼近理论在各种几何和非经典设置从多尺度系统，很好地适应目标的应用。第二个目标是研究从牛顿势的位移的谐波函数的近似，有针对性地应用于大地测量，地磁和物理。第三个目标是近似函数，这些函数在由光滑曲线或曲面分隔的空间域上是光滑的。这里的想法是使用自适应各向异性的多尺度膨胀的空间，使近似工具，以适应弯曲的奇点。该项目的核心目标是在各种几何和非经典环境中从框架和其他系统开发非线性n项近似，例如在球体，球，盒子和带权重的单形上，以及在李群和黎曼流形的背景下。所有这些设置覆盖的一般框架的Dirichlet空间的热核具有高斯边界。该方法的关键是使我们能够自由地处理（a）不同的几何，（B）紧的和非紧的空间，（c）具有非平凡权的空间，同时允许Besov和Triebel-Lizorkin空间的发展和框架分解具有完整的指标范围. 热核理论的发展和局域系统的非线性n项近似是这一理论的基本方面。这个项目的另一个目标是从牛顿势的位移的线性组合发展d维球上的调和函数的非线性近似。这包括近似率的完整表征和实现最佳近似率的有效算法的开发。各向异性现象出现在分析、偏微分方程和应用中的各种环境中。例如，函数在由光滑曲线或流形分隔的空间域上经常是非常光滑的。该项目旨在通过使用各向异性多尺度膨胀的框架来解决这种函数的奇异性，这种膨胀可以在任何水平和深度上从一点到另一点迅速变化。这种方法的主要特点是（i）快速构造最佳或接近最佳的扩张矩阵，从而获得最佳稀疏性的算法的发展，（ii）高度局部化的各向异性框架的构造及其在函数非线性逼近中的应用。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Product Besov and Triebel–Lizorkin Spaces with Application to Nonlinear Approximation

DOI：
10.1007/s00365-019-09490-1
发表时间：
2019-12
期刊：
Constructive Approximation
影响因子：
2.7
作者：
A. G. Georgiadis;G. Kyriazis;P. Petrushev
通讯作者：
A. G. Georgiadis;G. Kyriazis;P. Petrushev

Kernel and wavelet density estimators on manifolds and more general metric spaces

流形和更一般的度量空间上的核和小波密度估计器

DOI：
10.3150/19-bej1171
发表时间：
2020
期刊：
Bernoulli
影响因子：
1.5
作者：
Cleanthous, Galatia;Georgiadis, Athanasios G.;Kerkyacharian, Gerard;Petrushev, Pencho;Picard, Dominique
通讯作者：
Picard, Dominique

Nonlinear $n$-term approximation of harmonic functions from shifts of the Newtonian kernel

牛顿核位移的调和函数的非线性 $n$ 项近似