Calculation of Discrepancy Measures and Applications

TL;DR

This chapter surveys algorithms for computing geometric discrepancy measures of point sets, focusing on $L_2$-discrepancies and star discrepancy, and discusses three application examples.

Contribution

It provides a comprehensive overview of methods for calculating discrepancy measures and explores their applications, highlighting recent algorithmic developments.

Findings

Algorithms for $L_2$-discrepancies are well-established.

Calculating star discrepancy remains computationally challenging.

Three practical applications demonstrate the utility of discrepancy algorithms.

Abstract

In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the geometric discrepancy measures for which computation algorithms have been designed. In particular, we explain methods to determine $L_2$-discrepancies and approaches to tackle the inherently difficult problem to calculate the star discrepancy of given sample sets. We also discuss in more detail three applications of algorithms to approximate discrepancies.