Koksma–Hlawka Inequality

Abstract

AbstractThe Koksma–Hlawka inequality is a tight error bound on the approximation of an integral by the sample average of integrand values. The integration error is bounded by a product of two terms, the discrepancy of the sample points,\documentclass{minimal}\usepackage{amsmath}\begin{document}$D(\{\vec{x}_{i}\})$\end{document}, and the variation of the integrand,V(g). These two quantities measure the quality of the sample points and the roughness of the integrand, respectively. The Koksma–Hlawka inequality plays a key role in the development of quasi–Monte Carlo methods. Such methods replace simple random sample points by low discrepancy points. The Koksma–Hlawka inequality has also influenced the study of experimental design and led to the creation of UNIFORM DESIGNS.