Local and Nonlinear Kernel Approximation

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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.