Local and Nonlinear Kernel Approximation

HangelbroekDMS-1047694 本项目的目标是开发新的工具和算法，用于在欧氏流形和某些紧致流形上进行有效的核逼近。 这项工作包括两个重要方面。 第一个是开发计划来处理数据的非均匀排列（近似率由反映数据局部密度的参数控制）。 第二种方法是设计非线性方案，使用很少的核的线性组合来近似。开发的方案遵守主流近似理论的两个特征（对于基于核的近似方案来说，这些特征通常是难以捉摸的）：它们通过提供由近似的平滑度决定的收敛率来提供精确的近似，并且它们是通用的，通过在所有平滑度水平上处理近似。 使用核函数来处理离散的高维数据，到目前为止，是近似理论中一种成熟的方法。核函数因其在缺乏基本几何结构（如网格或三角剖分）的情况下进行近似的能力而受到特别重视。 在这一点上，有几个算法采用内核来处理几乎均匀采样的大型数据集。 然而，这种算法的近似能力--根据近似函数对近似函数的保真度来判断--很少被完全理解。此外，如何处理高度不均匀的数据（数据与大的差距，或与点合并）usingkernels的问题才刚刚开始得到解决。 该项目的一个重要目标是开发基于核的近似方法，该方法近似于高度非结构化的数据集，并且近似于高维数据集，计算开销很小。 另一个目标是获得对这种方法的近似能力的精确理解。 特别有趣的问题，有一些潜在的几何或代数结构被利用，例如，是酪蛋白问题在大地测量学，晶体学和分子生物学。

HangelbroekDMS-1047694 The goal of this project is to develop new tools andalgorithms for effective kernel approximation on Euclideandomains and certain compact manifolds. This work includes twoimportant aspects. The first is to develop schemes to treathighly nonuniform arrangements of data (with approximation ratescontrolled by a parameter reflecting the local density of thedata). The second is to devise nonlinear schemes thatapproximate using linear combinations of very few kernels. Schemes developed abide by two features of mainstreamapproximation theory (features that have generally been elusivefor kernel-based approximation schemes): they provideapproximation that is precise, by providing convergence ratesdictated by the smoothness of the approximand, and they areuniversal, by treating approximands at all levels of smoothness. The use of kernels to treat scattered, high-dimensional datais, by now, an established methodology in approximation theory. Kernels are especially prized for their ability to approximate inthe absence of underlying geometrical structures, like meshes ortriangulations. At this point there exist several algorithmsemploying kernels to treat large datasets that have been sampledalmost uniformly. However, the approximation power of suchalgorithms -- judged in terms of the fidelity of the approximantto the approximand -- is rarely completely understood. Furthermore, the question of how to treat highly nonuniform data(data with large gaps, or with points that coalesce) usingkernels is only beginning to be addressed. An important goal ofthis project is to develop kernel-based approximation methodsthat approximate from highly unstructured datasets and thatapproximate high-dimensional datasets with little computationaloverhead. Another goal is to acquire a precise understanding ofthe approximation power of such methods. Of particular interestare problems where there is some underlying geometric oralgebraic structure to be exploited, as, for example, is the casein problems in geodesy, crystallography, and molecular biology.