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

1714369 Petrushev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对目标应用程序的自然拓扑中的基本函数进行有效的表示和近似。捕获发生在不同尺度上的物理现象和数据结构需要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快速而准确地计算。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。第一个目标是发展适合于目标应用的多尺度系统在各种几何和非经典环境下的非线性逼近理论。第二个目标是研究牛顿势的位移对调和函数的逼近，并有针对性地应用于大地测量学、地磁学和物理学。第三个目标是逼近空间中由光滑曲线或曲面分隔的区域上光滑的函数。这里的想法是使用空间的自适应各向异性多尺度伸缩，这使得近似工具能够调整到曲线奇点。这个项目的一个核心目标是在各种几何和非经典环境中发展框架和其他系统的非线性n项逼近，例如在球、球、盒和带权的单形上，以及在李群和黎曼流形的背景下。所有这些设置都被具有高斯界的热核Dirichlet空间的一般框架所涵盖。这种方法的关键是给了我们处理(A)不同几何，(B)紧空间和非紧空间，(C)具有非平凡权的空间的自由，同时允许Besov空间和Triebel-Lizorkin空间的发展和框架分解具有完备的指数范围。基本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根据牛顿势的位移的线性组合，发展d维球上调和函数的非线性近似。这包括对近似率的完整描述，以及开发实现最佳近似率的有效算法。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，函数通常在由光滑曲线或流形分隔的空间中的区域上非常光滑。该项目旨在利用各向异性多尺度伸缩的框架来解决函数的这种奇异性，这种奇异性可以在任何水平和深度上从一个点到另一个点快速变化。该方法的主要内容是：(I)发展一种快速构造最佳或接近最佳伸缩矩阵的算法，以获得最佳稀疏性；(Ii)构造高度局部化的各向异性框架及其在函数的非线性逼近中的应用。

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.