The overall goal of this project is to develop new methods and tools for the analysis and synthesis of scattered data by means of radial basis functions, spherical basis functions, and tight spherical frames. The research will provide a foundation for the effectiveness of various radial basis function meshless methods for solving partial differential equations by estimating the errors made in using them; knowing these errors can then serve as a guide in creating and implementing algorithms based on radial basis and spherical basis functions, both for meshless methods for solving differential equations and for scattered data surface fitting. Tight, well-localized spherical frames are a newly discovered tool for dealing with scattered data on spheres, and are similar to frames used in connection with wavelet analysis. Very little is known about the specifics of these tight frames, and a second goal of this project is to develop them, and algorithms based on them, to efficiently process scattered data on spheres.
Problems involving analyzing and synthesizing data taken from scattered sites arise in diverse fields -- computer-aided design graphics, data mining, medical imaging, learning networks, geoscience, and many other areas. The methods under development in this project for solving partial differential equations will help in solving problems involving wave propagation, modeling fluid flow, and even tomography. The work on spherical basis functions and tight spherical frames will impact geoscience, especially in the case of fitting large scattered-data sets collected over the earth via satellites or by ground stations.
