Mathematical Sciences: Computational Aspects of MultivariateSplines

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This research effort is designed to go hand in hand with the theoretical work being carried out under NSF GRANT DMS-8602337, entitled "Theory and Applications of Multivariate Splines". The aim of this research is twofold. We propose 1) to translate the theoretical results obtained under the earlier grant into practical computational algorithms. In particular, software will be developed for triangulation and tesselation, for storing and evaluating piecewise polynomial surfaces efficiently, for contouring, for surface-surface intersection, for interpolation (with and without shape control), for quasi-interpolation, for fitting surfaces on surfaces, for computer-aided design schemes, for least squares fitting of scatted data, for evaluation of simplex splines and for solving boundary-value problems. Secondly, we propose 2) to develop computational tools which can be used to support our theoretical work on the dimension, the construction of locally supported bases, and the poisedness of interpolation problems for multivariate splines.