Triangulations by Reflections with Applications to Approximation

Abstract

Triangulations of subsets of ℝⁿ, or simplicial coverings of subsets of ℝⁿ may be used as a helpful device in such numerical approximation tasks as numerical integration [14], [15], piecewise approximation of functions, and finite element methods [17], [18]. Triangulations also play a crucial role in combinatorial algorithms for the approximation of fixed points of mappings [3], [4], [5], [9], [13]. Recently the authors have shown [1] that triangulations of ℝⁿ can be generated by performing reflections of vertices of simplices across edges in a certain prescribed way. Pivoting between simplices by reflections permits the generation of sequences of simplices with elementary programming, minimal storage, and versatile starting. Our objective here is to develop algorithms for sequential triangulations which may be used in the above mentioned applications.

This paper was written while the first author was a guest of the Sonderforschungsbereich 72 ‘Approximation und Optimierung’ at University in Bonn