Intersplines: A New Approach to Globally Optimal Multivariate Splines Using Interval Analysis

In science and engineering there often is a need for the approximation of scattered multi-dimensional data. A class of powerful scattered data approximators are the multivariate simplex B-splines. Multivariate simplex B-splines consist of Bernstein basis polynomials that are defined on a geometrical structure called a triangulation. Multivariate simplex B-splines have a number of advantages over other scattered data approximators, like neural networks and kernel methods. Firstly, the multivariate simplex B-splines have an arbitrarily high approximation power. Secondly, the simplex spline models are parametric models, which allows for efficient approximation of arbitrarily large datasets. Finally, the local basis property of the simplex B-splines results in sparse solution and evaluation systems. Until now, the disadvantage of using simplex B-splines for scattered data approximation is that a triangulation must be defined a-priori ∗Submitted: July 22, 2011; Revised: June 7, 2012; Accepted: July 20, 2012.