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

TL;DR

This paper develops a new method to generate uniformly distributed sequences in the orthogonal group and Grassmannian manifold, enhancing quasi-Monte Carlo techniques for numerical integration and geometric approximation.

Contribution

It introduces a novel construction of uniformly distributed sequences in O(n) and G(n,k), extending quasi-Monte Carlo methods to new geometric settings.

Findings

Sequence compares favorably with random methods

Effective for integral-geometric formula approximation

Motivates future research in geometric quasi-Monte Carlo

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.