Theory and Applications of Localized Kernel Bases to Meshfree Methods

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The need for analyzing and modeling data taken from scattered, irregularly placed sites arises frequently in diverse fields: atmospheric science, artificial intelligence, computer-aided design graphics, data mining, medical imaging, learning networks, and many other areas. For example, weather prediction or climate modeling is based on geophysical data collected at scattered sites, by sensors on satellites, ground stations, or stations at sea. Carrying out such tasks presents difficulties for traditional methods, which are based on collecting data at uniformly placed sites or which require constructing "meshes" (think wire fence) that must be carefully tailored to deal with the data sites involved. Newer methods, the so-called kernel methods, are meshfree and can handle scattered data. The investigators further develop the theory of kernel methods, on the basis of which algorithms can be developed that are easier to use, faster, less expensive to implement, and capable of handling data from a hundred thousand or more sites.Scattered data problems present a challenge for any method, including the traditional kernel-type algorithms based on translates of one (conditionally) positive definite function. Scattered data occur naturally in meshfree methods, machine learning, neural nets, and other situations. Dealing with such data, ideally, requires local, stable bases, preconditioners, and other similar tools. In recent work on the sphere and rotations in space where no boundary is present, the investigators have developed novel bases related to certain classes of kernels. For problems where boundaries are inherent, their bases need further development. One key part of this project involves a novel idea of extrapolating data slightly beyond the boundary of a compact domain to enhance the approximation power of the method. Another key area of investigation involves local approximation orders for data that is far more general than the typical quasi-uniform assumptions.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.