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

1714369Petrushev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中有效地表示和近似底层函数。捕获在不同尺度上发生的物理现象和数据结构需要从相对于应用领域的本地支持的多尺度系统中进行近似。此外，这些近似方法应能适应快速和精确的计算。因此，该项目旨在增加我们对非线性近似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近似理论在各种几何和非经典设置从多尺度系统，很好地适应目标应用。第二个目标是研究牛顿势的位移对谐波函数的近似，并有针对性地应用于大地测量学、地磁和物理学。第三个目标是在由光滑曲线或曲面分隔的空间域上近似光滑的函数。这里的想法是使用空间的自适应各向异性多尺度膨胀，这使得近似工具能够调整到弯曲的奇点。该项目的核心目标是在各种几何和非经典环境下，如球体、球、盒子和带权的单纯形，以及在李群和黎曼流形的背景下，从框架和其他系统中发展非线性n项逼近。所有这些设置都包含在具有高斯边界的热核狄利克雷空间的一般框架中。该方法的关键在于给予我们处理(a)不同几何形状，(b)紧致和非紧致空间，(c)非平凡权值空间的自由，同时允许Besov和triiebel - lizorkin空间的发展和框架分解。基础热核理论和局域系统非线性n项近似的发展是该理论的基本方面。这个项目的另一个目标是从牛顿势的位移的线性组合中发展出d维球上谐波函数的非线性逼近。这包括逼近率的完整表征和实现最佳逼近率的有效算法的发展。各向异性现象出现在分析、偏微分方程和应用的各种环境中。例如，函数在由光滑曲线或流形分隔的空间域上通常是非常光滑的。该项目旨在利用各向异性多尺度膨胀的框架来解决这种函数的奇异性，它可以在任何层次和深度上从点到点快速变化。该方法的主要内容是：(i)开发一种快速构建最佳或接近最佳扩张矩阵的算法，从而获得最佳稀疏性；（ii）构建高度局部化的各向异性框架及其在函数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.