Uniformly distributed sequences in the orthogonal group and on the Grassmannian manifold

Abstract

Quasi-Monte Carlo methods replaced classical Monte Carlo methods in many areas of numerical analysis over the last decades. The purpose of this paper is to extend quasi-Monte Carlo methods into a new direction. We construct and implement a uniformly distributed sequence in the orthogonal group O(n). From this sequence we obtain a uniformly distributed sequence on the Grassmannian manifold $G(n,k)$, which we use to approximate integral-geometric formulas. We show that our algorithm compares well with classical random constructions and, thus, motivate various directions for future research.