The goal of this project is to develop new tools and algorithms for effective kernel approximation on Euclidean domains and certain compact manifolds. This work includes two important aspects. The first is to develop schemes to treat highly nonuniform arrangements of data (with approximation rates controlled by a parameter reflecting the local density of the data). The second is to devise nonlinear schemes that approximate using linear combinations of very few kernels. Schemes developed abide by two features of mainstream approximation theory (features that have generally been elusive for kernel-based approximation schemes): they provide approximation that is precise, by providing convergence rates dictated by the smoothness of the approximand, and they are universal, by treating approximands at all levels of smoothness.
The use of kernels to treat scattered, high-dimensional data is, by now, an established methodology in approximation theory. Kernels are especially prized for their ability to approximate in the absence of underlying geometrical structures, like meshes or triangulations. At this point there exist several algorithms employing kernels to treat large datasets that have been sampled almost uniformly. However, the approximation power of such algorithms -- judged in terms of the fidelity of the approximant to 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) using kernels is only beginning to be addressed. An important goal of this project is to develop kernel-based approximation methods that approximate from highly unstructured datasets and that approximate high-dimensional datasets with little computational overhead. Another goal is to acquire a precise understanding of the approximation power of such methods. Of particular interest are problems where there is some underlying geometric or algebraic structure to be exploited, as, for example, is the case in problems in geodesy, crystallography, and molecular biology.
