The main object of this project is to develop new methods and tools for the analysis and synthesis of scattered data by means of radial and related basis functions. Very recently, the investigators Ward and Narcowich and a collaborator introduced a new tool for fitting a divergence-free vector field tangent to a 2D orientable surface contained in 3D to samples of such a field taken at scattered sites on the surface. The method, which involves a kernel constructed from radial basis functions (RBFs), has applications to problems in geophysics -- for instance, the nonlinear flow of an incompressible fluid in a single hydrostatic atmospheric layer. Mathematically, incompressibility translates to the velocity having vanishing surface divergence. Fitting divergence-free tangent vector fields to data taken in these cases would help in modeling the incompressible velocity fields involved. In addition to handling scattered data, it has the advantage of avoiding special treatment for the poles. The investigators study the approximation properties of this tool, with a view towards estimates on its data-fitting errors as well how robust it is. For certain cases, numerical experiments suggest both a good fit of data and robustness. In addition, unstructured scattered data -- i.e., clumped, or clustered, or nonuniformly distributed data -- present problems for any method, including the traditional RBF-type algorithms based on translates of one conditionally positive definite function. These clustered data occur naturally in learning theory, neural nets, meshless methods, and other situations. To deal with clustered data, the investigators initiate the study of the construction of new bases, preconditioners, and in general a new RBF-type paradigm.
Problems that involve analyzing data taken from scattered sites -- that is, irregularly placed sites -- in space or on the surface of the earth arise frequently in diverse fields: computer-aided design graphics, data mining, medical imaging, learning networks, and geoscience, in addition to many other areas. Such problems present difficulties for traditional methods, which are based on collecting data at uniformly placed sites. For example, weather prediction is based on a mathematical model where the earth's surface is assumed to be a surface of a sphere. The investigators develop new methods and tools to help in analyzing such phenomena via so-called radial basis function methods.
