Highly Nonlinear Approximation: Theory and Algorithms

Pi：Pencho Petrushev，南卡罗来纳大学DMS-0200665=摘要该项目将集中在大型冗余系统的非线性n项逼近中的具体问题，这通常被称为高度非线性逼近。冗余系统已经在信号和图像处理以及总体上对近似理论产生了严重影响。研究界目前正在开发两种形式的冗余系统。第一类是小波包和余弦包等基函数库。第二个主要推动力是研究Gabor函数和神经网络等词典。我们的研究重点是不同性质的词典，它们分为两类：(I)由无限光滑的快速衰减函数的移位和展开组成的词典，其中包括高斯、径向部分分数或由这些函数的某些变换组成的更一般的各向异性系统。(Ii)由空间的多层并元划分、多层嵌套三角剖分或单纯形划分生成的分段多项式的集合。该项目的主要目标是了解非线性和高度非线性的过程。特别是，我们必须发展对光滑性条件(由特定空间指定)的性质的理解，这些条件支配着重要度量集合中的逼近速度。该项目的第二个主要工作是开发以基本分析为基础、能够达到最佳逼近(即压缩)的速度和实用的算法。该项目正在积极协调其与合作伙伴的研究，这些合作伙伴涉及分析和呈现广泛应用领域的数据，包括数字高程图和来自地理信息系统(GIS)的相关图像、信号和图像分析、计算机辅助几何设计(CAGD)、图形绘制和大规模数值模拟的可视化。在每一种情况下，目标都是在规定的词典的背景下分析复杂的数据和信号，并将它们分解成对被调查的物理系统来说是自然的基本元素。与基本元素相关联的幅度，无论它们是小波、多分辨率冗余系统还是各向异性系统，都提供关于数据的固有信息，并且理想地导致信号类别(数据)的最大熵编码。数据的这种编码表示对于存储、传输、快速查询、可视显示、关联或与来自其他模式的数据进行配准是重要的。这个项目的主要动机是为这些调查的基本理论基础做出实质性贡献。

PI: Pencho Petrushev, University of South CarolinaDMS-0200665 ========ABSTRACT The project will be focused on specific problems in nonlinear n-term approximation from large redundant systems, which is commonly referred to as highly nonlinear approximation. Redundant systems have already had a serious impact in signal and image processing and upon Approximation Theory in general. The research community is currently developing two forms of redundant systems. The first deals with libraries of bases such as the wavelet packets and cosine packets. The second major thrust has been to study dictionaries such as the Gabor functions and neural networks. The focus of our investigations is toward dictionaries of a different nature, which fall in two categories:(i) Dictionaries consisting of shifts and dilates of infinitely smooth, rapidly decaying functions which include the Gaussian, radial partial fractions, or more general anisotropic systems consisting of certain transformations of such functions. (ii) Collections of piecewise polynomials generated by multilevel dyadic partitions of the pace, multilevel nested triangulations, or simplex partitions. The primary goal of project is to understand nonlinear and highly nonlinear processes. In particular, we must develop an understanding of the nature of the smoothness conditions (specified by specific spaces) which govern the rate of approximation in a collection of important metrics. The second major effort of the project is to develop algorithms which are grounded in fundamental analysis, capable of achieving the rate of the best approximation (i.e. compression), and are practical to implement.This project is actively coordinating its investigations with collaborators involved in the analysis and presentation of data for a wide range of application fields, including digital elevation maps and associated imagery arising from geographical information systems (GIS), signal and image analysis, computer aided geometric design (CAGD), graphical rendering and visualization of large scale numerical simulations. In each case, the objective is to analyze complicated data and signals in the context of prescribed dictionaries and to decompose them into basic elements which are natural for the physical system under investigation. The amplitudes associated with the base elements, whether they be wavelets, a multiresolution redundant system, or anisotropic systems, provide inherent information about the data and ideally lead to the maximal entropy encoding of classes of signals (data). This encoded representation of data is important for the storage, transmission, fast query, visual display, correlation, or registration against data from other modalities. The primary motivation of this project is to make substantial contributions to the underlying theoretical foundation for these investigations.