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Calculation of Discrepancy Measures and Applications

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A Panorama of Discrepancy Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2107))

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.

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Notes

  1. 1.

    BMO stands for “bounded mean oscillation”.

  2. 2.

    NP stands for “non-deterministic polynomial time”.

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Acknowledgements

The authors would like to thank Sergei Kucherenko, Shu Tezuka, Tony Warnock, Greg Wasilkowski, Peter Winker, and an anonymous referee for their valuable comments.

Carola Doerr is supported by a Feodor Lynen postdoctoral research fellowship of the Alexander von Humboldt Foundation and by the Agence Nationale de la Recherche under the project ANR-09-JCJC-0067-01.

The work of Michael Gnewuch was supported by the German Science Foundation DFG under grant GN-91/3 and the Australian Research Council ARC.

The work of Magnus Wahlström was supported by the German Science Foundation DFG via its priority program “SPP 1307: Algorithm Engineering” under grant DO 749/4-1.

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Authors and Affiliations

  1. Université Pierre et Marie Curie - Paris 6, LIP6, équipe RO, Paris, France

    Carola Doerr

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Carola Doerr & Magnus Wahlström

  3. Mathematisches Seminar, Kiel University, Kiel, Germany

    Michael Gnewuch

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  1. Carola Doerr
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  2. Michael Gnewuch
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  3. Magnus Wahlström
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Correspondence to Michael Gnewuch .

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  1. Department of Mathematics, Macquarie University, Sydney, New South Wales, Australia

    William Chen

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Anand Srivastav

  3. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milano, Italy

    Giancarlo Travaglini

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Doerr, C., Gnewuch, M., Wahlström, M. (2014). Calculation of Discrepancy Measures and Applications. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_10

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