RUI: Modelling of Scattered Data on Manifolds

California State L A University Auxiliary Services Inc., Los Angeles CA

Investigators

Abstract

DMS Award Abstract Award #: 0204704 PI: Mhaskar, Hrushikesh Institution: California State University Los Angeles Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: RUI: Modelling of Scattered Data on Manifolds The proposer will continue his research on the modelling and analysis of scattered data on the sphere and other manifolds. For the analysis of data, he will develop frames having small or compact supports, and having dual frames with other desirable properties. The proposer will study the optimal representation of functions on the manifolds using finitely many bits, rather than real coefficients, and develop algorithms for these representations based on scattered data. An important new mathematical tool in this study will be quadrature formulas with positive weights based on data near (rather than on) the manifolds, and local quadrature formulas based on data near parts of the manifolds. The research is expected to have applications in the areas of signal processing, analysis of satellite data, and the study of development of tumors. The problem of approximation of functions on the sphere is a very old one; among the early workers in this area was Gauss. It arises naturally in geodesic studies, for example, the study of the environment, and variations in the gravitational and electromagnetic fields around the earth. The data taken from a satellite is most naturally modelled as a function on the sphere. Additional applications arise in mathematical biology, where the growth of a tumor is modelled by a system of differential equations satisfied by functions on a sphere, which need to be approximated efficiently and accurately in real time. Many recent applications require a study of functions on slight perturbations of the sphere. For example, the data from the satellite may be taken not from the surface of the earth itself (which is not perfectly spherical either), but from different heights above the surface. Similarly, in biological applications, a tumor which may be initially assumed spherical, no longer remains so after its evolution. The proposer will study the analysis and modelling of data collected at arbitrary sites on a nearly spherical manifold, using ideas from wavelet analysis and metric entropy. Date: June 18, 2002