Localized Kernel Bases with Application to Meshless Methods

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Problems involving the analysis and synthesis of data taken from scattered sites in space or on surfaces arise in diverse fields -- computer-aided design graphics, data mining, medical imaging, learning networks, geoscience, and many other areas. This award will support the development of new methods and tools for attacking the analysis and synthesis of scattered data -- i.e. data collected from non-uniformly distributed sites -- by means of kernel methods. The investigators supported by this awards recently discovered highly localized bases derived from special kernels on manifolds, and this breakthrough will play a key role both in the problems proposed and in their approach to investigating them. The problem of finding a good, stable basis for an approximation space made from kernels is closely connected with determining well-localized bases at low computational cost. One of the major difficulties in dealing with kernel interpolation or least squares approximation with the ``standard'' kernel basis is that collocation matrices are full and frequently ill conditioned. Recent work by the investigators and collaborators showed that Lagrange-type interpolating functions associated with certain kernels are exponentially localized about their centers and, from numerical experiments, appear to be computationally inexpensive. This award will support exploring the full potential of these newly discovered basis functions.The need for analyzing and modeling data taken from scattered, irregularly placed sites arises frequently in diverse fields: computer-aided design graphics, data mining, medical imaging, learning networks, and geoscience, in addition to 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, do not require such meshes and can handle scattered data. This award will further the development of these kernel methods, making them easy to use, faster, less expensive to implement, and capable of handling data from a hundred thousand or more sites. It will provide support for graduate students, who will be trained in both the theoretical and the applied aspects of using and developing these methods.