## Günter Rote and Robert Franz Tichy:

# Quasi-Monte-Carlo methods and the dispersion of point sequences

*Mathematical and Computer Modelling 23 (1996), 9-23, (Zbl 855.11041).
*
### Abstract

Quasi-Monte-Carlo methods are well-known for solving different problems of
numerical analysis such as integration, optimization, etc. The error estimates
for global optimization depend on the dispersion of the point sequence with
respect to balls.
In general, the dispersion of a point set with respect to various classes of
range spaces, like balls, squares, triangles, axis-parallel and arbitrary
rectangles, spherical caps and slices, is the area of the largest empty
range, and it is a measure for the distribution of the points. The main
purpose of our paper is to give a survey about this topic, including some
folklore results. Furthermore, we prove several properties of the dispersion,
generalizing investigations of Niederreiter and others concerning balls. For
several well-known uniformly distributed point sets we estimate the dispersion
with respect to triangles, and we also compare them computationally. For the
dispersion with respect to spherical slices we mention an application to the
polygonal approximation of curves in space.

Last update: August 16, 2004.