Local and Nonlinear Kernel Approximation

本项目的目标是开发新的工具和算法，在欧几里得域和某些紧流形上进行有效的核逼近。这项工作包括两个重要方面。第一个是开发方案来处理高度不均匀的数据排列（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，利用极少核的线性组合进行近似。所开发的方案遵循主流近似理论的两个特征（对于基于核的近似方案来说通常是难以捉摸的特征）：它们通过提供由近似的平滑性决定的收敛率来提供精确的近似，并且它们是通用的，通过在所有平滑程度上处理近似。利用核函数来处理分散的高维数据，目前已成为近似理论中的一种确定的方法。核函数在没有底层几何结构（如网格或三角剖分）的情况下进行近似的能力尤其受到重视。在这一点上，有几种算法使用核来处理几乎一致采样的大型数据集。然而，这种算法的近似能力——根据近似值对近似值的保真度来判断——很少被完全理解。此外，如何使用核处理高度不一致的数据（具有大间隙的数据，或具有合并点的数据）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，该方法从高度非结构化的数据集进行近似，并且以很少的计算开销近似高维数据集。另一个目标是获得对这些方法的近似能力的精确理解。特别有趣的是，有一些潜在的几何或代数结构有待开发的问题，例如，大地测量学、晶体学和分子生物学中的酪蛋白问题。

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.