New Directions in Mesh-Free Approximation with Localizable Kernels

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Kernels and radial basis functions are valuable mathematical tools that are commonly used to treat large and challenging computational problems. They enjoy widespread use in a variety of branches of science and engineering, like computational chemistry, machine learning, computer graphics, and geosciences, where researchers wish to approximate, model, or extract information from geometrically complicated data sets. This project is devoted to improving the stability and speed of algorithms which use kernels, to apply these algorithms to a number of important scientific problems, and to dramatically extend the approximation power of kernels beyond the current state of the art by developing a multi-scale, anisotropic theory of kernel approximation. Results will benefit those scientific fields that use kernels as a computational tool. Throughout this project, the investigator will continue his own research into these problems, while mentoring graduate students by involving them in theoretical and computational aspects.This project involves exploiting local structures within underlying kernel spaces to study linear and non-linear kernel approximation with non-uniform geometry and multi-scale, anisotropic kernels. One aspect of this project treats kernels of finite smoothness for which there exist highly localized Riesz bases. Such bases confer a number of well-understood analytic benefits, but their true promise is in improving kernel based algorithms; the investigator will employ these local structures to stabilize and speed up computation, and to develop novel mesh-free numerical methods for scattered data and partial differential equations. Another aspect involves Gaussians and other infinitely smooth kernels having extreme localization in both space and frequency. The investigator will study and develop nonlinear approximation methods that use multi-scale, anisotropic operations to treat functions which are sparse in a variety of representation systems, employing techniques from harmonic analysis and statistics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.