Local and Nonlinear Kernel Approximation

Hangelbroek DMS-1047694这个项目的目标是开发新的工具和算法来有效地逼近欧几里得范数和某些紧致流形。这项工作包括两个重要方面。第一种是开发数据的高度非均匀排列方案(近似率由反映数据局部密度的参数控制)。第二种是设计使用极少数核的线性组合近似的非线性方案。所开发的方案遵循主流逼近理论的两个特征(基于核的逼近方案通常难以捉摸的特征)：它们通过提供由逼近的光滑度决定的收敛速率来提供精确的逼近，并且它们是通用的，通过在所有光滑度级别上处理逼近。到目前为止，使用核来处理分散的高维数据是近似理论中一种既定的方法。核函数因其在没有基本几何结构(如网格或三角剖分)的情况下进行近似的能力而特别受到重视。在这一点上，有几种算法使用内核来处理几乎均匀采样的大数据集。然而，这种算法的逼近能力--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(具有较大间隙的数据或具有聚合点的数据)的问题才刚刚开始解决。这个项目的一个重要目标是开发基于核的近似方法，该方法可以从高度非结构化的数据集进行近似，并且以很小的计算开销近似高维数据集。另一个目标是对这些方法的逼近能力有一个精确的理解。特别令人感兴趣的是有一些潜在的几何或代数结构需要利用的问题，例如，测地学、结晶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.