Multi-Dimensional Polynomial Interpolation on Arbitrary Nodes
Dongbin Xiu
Department of Mathematics
Purdue University
Polynomial interpolation is well understood on the real line. And many techniques in multi-dimensional space employs the one-dimensional methods and fill up the space using certain tensor product rule. The typically results in fast growth of the total number of interpolation nodes and certain fixed geometrical structure of the nodal sets. In practice, one often have function data from nodal locations that are not by "mathematical design". These nodes typically possess no desirable structure and their total number can be arbitrary.
In this talk, we present a mathematical framework for conducting polynomial interpolation in multiple dimensions using arbitrary set of unstructured nodes. The resulting method,least orthogonal interpolation, is rigorous and has a straightforward numerical implementation. It can faithfully interpolate any function data on any nodal sets, even on those that are considered degenerate by the traditional methods.