New Directions in Scattered Data Analysis via Radial and Related Basis Functions with Applications

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Error estimates for both interpolation and approximation are central in developing rigorous algorithms or numerical methods in any application. A major difficulty in scattered-data problems has been that the class of data-generating functions for which the methods are known to converge is much smaller than what one encounters in practice. This is problematic in numerically solving partial differential equations via Radial Basis Function collocation, since target functions need to be smoother than the basis functions, which is a major restriction in hyperbolic problems. One of the goals of this project is to obtain error estimates for a greatly expanded class of target functions. In our very recent work on interpolation via a restricted class of Spherical Basis Functions on the sphere, such estimates were obtained. This provides hope that the broader goal is attainable. Another important goal is to develop and implement rigorous, computationally efficient algorithms for numerical partial differential equation problems, neural networks, and problems from the geosciences requiring scattered-data surface fitting on the sphere.The investigation of scattered-data modeling is of great potential importance for the understanding of earth based phenomena of every kind. Fitting surfaces to meteorological or geophysical data collected via satellites or ground stations is a good example of such an application. Spherical basis functions (SBFs) have been used in such problems. Radial and periodic basis functions have been extensively employed in a variety of neural networks, including architectures used for direction-finding via phased-array radar. Very recently, they have been employed in grid-free numerical methods for solving partial differential equations, and to do computer graphics and computer aided design problems. Calculations assocaited with these are time consuming. Recent advances have ameliorated these difficulties and the work under this project will continue to improve efficiency.