Problems involving the approximation and representation of data taken from scattered sites in space, from surfaces, or from more complicated structures arise in diverse scientific fields. Kernel methods are valued for their ability to treat such unstructured data. They are also important examples of meshless methods, computational tools that function successfully without the need of accompanying structures like grids or triangulations. The principal investigator plans to study analytic and computational properties of scalable localized bases (a powerful new methodology for working with kernels) and to use these bases to address some fundamental challenges in kernel based approximation. The investigator expects to generate algorithms that will be of use to scientists and engineers who work with unstructured data and large-scale datasets. Throughout the project, the investigator will mentor graduate students by involving them in both the theoretical and the applied aspects of this research.
Unlike other computational tools (like wavelets, splines and piecewise polynomial finite elements), kernel methods do not explicitly involve a scaling operation: as data becomes more dense, the kernel is not required to be dilated. This allows kernels to produce approximates from complex geometrical configurations where a single scale is not apparent. However, it often leads to problems involving large, poorly conditioned linear systems, the solution of which is both slow and unstable. It has been demonstrated that in many cases the underlying function spaces possess stable, highly localized bases that can be constructed rapidly. There is compelling evidence that near linear processing of many computational problems is possible, and fast algorithms may exist to treat many of the basic operations involving kernels. The proposed research seeks to generate scalable bases in new settings (i.e., for new kernels and on new manifolds), to understand and overcome fundamental problems stemming from boundary effects and highly non-uniform arrangements of data, and ultimately to implement parallelized, fast algorithms for kernel approximation with these bases.
