Representation and approximation of functions in nonclassical and anisotropic settings with applications

该项目的中心是在各种非经典和/或各向异性的拓扑设置，如李群和黎曼流形，并在空间的各向异性膨胀的框架中的多尺度表示系统的发展。此外，它将开发复杂的新方法，近似和数值计算使用这些系统。该项目位于计算谐波分析，频谱分解，正交多项式，非线性近似和数值分析之间的接口。它分为两个主要的调查方向，有几个具体的目标。该项目的第一个研究目标是在严格局部正则Dirichlet空间的一般背景下，利用加倍测度和局部尺度不变的Poincare不等式，构造出具有近指数空间局部化元素的框架，从而得到具有小时间高斯界和保持器连续性的马尔可夫热核.所提出的方法的关键点是能够处理（a）不同的几何，（B）紧和非紧空间，（c）具有非平凡权的空间，并且同时允许Besov和具有完整索引范围的其他空间的框架分解。这将有助于在李群或具有多项式体积增长的齐次空间，具有Ricci曲率下界的黎曼流形和其他新设置的背景下发展良好的局部化框架。在单纯形和图上的框架的发展是这个项目的另一个目标。该项目的第二个核心目标是在空间各向异性膨胀的框架内开发自适应表示。各向异性现象出现在分析、偏微分方程和应用中的各种情况下。例如，函数在由光滑曲线或流形分隔的Rd的子域上经常是非常光滑的。该项目旨在通过利用d维空间或其子域的各向异性多尺度膨胀的框架来解决函数的这种奇异性（以及更一般的奇异行为），这些空间或子域可以在任何水平和深度上从点到点快速变化。这种方法的主要思路是（i）快速构造最佳或接近最佳的膨胀矩阵的算法的发展，从而导致最佳稀疏性，（ii）高度局部化的各向异性框架的构造及其用于表示和逼近函数。许多科学领域需要在目标应用的自然拓扑中有效地表示底层函数。捕获发生在不同尺度的物理现象需要本地支持的多尺度系统相对于应用领域。此外，这些系统应该能够进行快速和准确的计算。这种系统（称为针状物）最近由研究者和他的合作者为球体和球开发。针是一个完整的重新思考的数据表示的背景下，经典的正交和频谱表示，并打破新的概念和实际的地面，远远超出了传统的多尺度的想法，如小波。球形针状体已经在宇宙学/天体物理学中对宇宙微波背景辐射数据的统计研究产生了重大影响。该奖项将支持图像和数据处理技术的发展，这将导致在新的数学设置中产生许多新颖的表示系统，从而可以处理新的数据结构。它还将通过开发创新的自适应方法来增强我们对复杂过程的基本理解，这些方法用于高效（稀疏）表示和近似具有跳跃或其他沿沿着曲线或曲面的急剧过渡的几何对象。它有可能影响许多领域，从图像处理和边缘检测到地球位势、海洋学和大气建模，以及物理学和宇宙学。

This project centers on the development of multiscale representation systems in various nonclassical and/or anisotropic topological settings such as on Lie groups and Riemannian manifolds and in the framework of anisotropic dilations of the space. Furthermore, it will develop sophisticated new methods for approximation and numerical computation using these systems. The project lies at the interface between computational harmonic analysis, spectral decompositions, orthogonal polynomials, nonlinear approximation and numerical analysis. It is organized into two main directions of investigation with several specific goals. The first research objective of the project is to develop frames with elements of nearly exponential space localization in the general setting of strictly local regular Dirichlet spaces with doubling measure and local scale-invariant Poincare inequality which lead to a Markovian heat kernel with small time Gaussian bounds and Holder continuity. The key point of the proposed approach is to be able to deal with (a) different geometries, (b) compact and noncompact spaces, and (c) spaces with nontrivial weights, and at the same time to allow for the frame decomposition of Besov and other spaces with complete range of indices. This will facilitate the development of well localized frames in the context of Lie groups or homogeneous spaces with polynomial volume growth, Riemannian manifolds with Ricci curvature bounded from below and other new settings. The development of frames on the simplex and on graphs is another aim of this project. The second core objective of this project is the development of adaptive representations in the framework of anisotropic dilations of the space. Anisotropic phenomena appear in various contexts in analysis, PDEs and in applications. For instance, functions are frequently very smooth on subdomains of Rd separated by smooth curves or manifolds. This project aims at resolving this kind of singularities of functions (and more general singular behaviors) by utilizing the framework of anisotropic multiscale dilations of d-dimensional space or its subdomain, 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 algorithms for rapid construction of best or near best dilation matrices leading to optimal sparsity, (ii) the construction of highly localized anisotropic frames and their utilization to representation and approximation of functions.Many scientific areas require efficient representation of the underlying functions in the natural topology of the targeted application. The capturing of physical phenomena occurring at various scales requires locally supported multiscale systems relative to the application domains. Moreover, these systems should be amenable to fast and accurate computation. Such systems (called needlets) have been recently developed by the investigator and his collaborators for the sphere and the ball. The needlets are the outcome of a complete rethinking of data representations in the context of classical orthogonal and spectral representations and break new conceptual and practical ground, going far beyond traditional multiscale ideas like wavelets. Spherical needlets have already had a significant impact in cosmology/astrophysics for the statistical study of the cosmic microwave background radiation data. This award will support the development of image and data processing techniques, which will lead to much novel representation systems in new mathematical settings, allowing the treatment of new data structures. It will also enhance our fundamental understanding of complicated processes through the development of innovative adaptive methods for efficient (sparse) representation and approximation of geometrical objects that have jumps or other sharp transitions along curves or surfaces. It has the potential to impact many areas ranging from image processing and edge detection to geopotential, oceanographic and atmospheric modeling, and to physics and cosmology.