Have a personal or library account? Click to login
On the Expected ℒ2–Discrepancy of Jittered Sampling Cover

On the Expected ℒ2–Discrepancy of Jittered Sampling

Open Access
|Aug 2023

References

  1. AISTLEITNER C.—PAUSINGER, F.—TICHY, R. F.—SVANE, A. M.: On functions of bounded variation, Math. Proc. Cambridge Philos. Soc. 162 (2017), 405–418.
  2. DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge, 2010.
  3. DOERR, B.: A sharp discrepancy bound for jittered sampling,Math. Comp. 91 (2022), 1871–1892.
  4. DOERR, C.—GNEWUCH, M.—WAHLSTRÖM, M.: Calculation of Discrepancy Measures and Applications In: Panorama of Discrepancy Theory, (W. W. L. Chen, A. Srivastav, G. Travaglini, eds.), Lecture Notes in Mathematics Vol. 2107, Springer-Verlag (2014). pp. 621–678.
  5. FRANK, K.—HEINRICH, S.: Computing discrepancies of Smolyak quadrature rules, J. Complexity 12 (1996), 287–314.
  6. GNEWUCH, M.—SRIVASTAV, A.—WINZEN C.: Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems,J.Complexity 25 (2009), 115–127.
  7. HEINRICH, S.: Efficient algorithms for computing the L2 discrepancy, Math. Comput. 65 (1996), 1621–1633.
  8. HICKERNELL, F. J.: A generalized discrepancy and quadrature error bound, Math. Comp. 67 (1998), 299–322.
  9. HICKERNELL, F. J.: The mean square discrepancy of randomized nets,ACM Trans. on Modeling and Computer Simulation. 6 (1996), no. 4, 274–296.
  10. HLAWKA, E.: Funkionen von Beschränkter Variation in der Theorie der Gleichverteilung, Ann. Mat Pura Appl. 54 (1961), no. 4, 325–333.
  11. JENSEN, J. L. W. V.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes,Acta Math. 30 (1906), no. 1, 175–193.
  12. KOKSMA, J. F.: Some integrals in the theory of uniform distribution modulo 1,Mathematica, B (Zutphen) 11 (1942), 49–52. (In Dutch)
  13. KOKSMA, J. F.: A general theorem from the theory of uniform distribution modulo 1, Mathematica, Zutphen B. 11 (1942), 7–11. (In Dutch)
  14. KRITZINGER, R.: An exact formula for the L2-discrepancy of the symmetrized Hammersley point set, Math. Comput. Simulation 143 (2018), 3–13.
  15. KIDERLEN, M.—PAUSINGER, F.: Discrepancy of stratified samples from partitions of the unit cube, Monatsh. Math. 195 (2022), 267–306.
  16. KIDERLEN, M., PAUSINGER, F.: On a partition with a lower expected L2−discrepancy than classical jittered sampling,J. Complexity 70 (2022), Article ID 101616, 13 pp.
  17. LEOBACHER, G.—PILLICHSHAMMER, F.: Bounds for the weighted Lp-discrepancy and tractability of integration,J. Complexity, 19 (2003), 539–547 .
  18. LEOBACHER, G.—PILLICHSHAMMER, F.: Introduction to Quasi-Monte Carlo Integration and Applications.Birkh¨auser/springer, Cham, 2014.
  19. LEMIEUX, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer Science + Business Media, Springer, New York, 2009.
  20. MATOUŠEK, J.: On the L2-discrepancy for anchored boxes,J.Complexity 14 (1998), 527–556.
  21. MATOUŠEK, J.: Geometric Discrepancy. Springer-Verlag, Berlin, 1999.
  22. NIEDERREITER, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992.
  23. PAUSINGER, F.—STEINERBERGER, S.: On the discrepancy of jittered sampling, J. Complexity 33 (2016), 199–216.
  24. PAUSINGER, F.—SVANE, A. M.: A Koksma-Hlawka inequality for general discrepancy systems,J. Complexity 31 (2015), 773–797.
  25. WARNOCK, T. T.: Computational investigations of low discrepancy point sets, In: Analysis, Proc. Sympos. Univ. Montreal 1971 (S. K. Zaremba, ed.), Appl. Number Theory numer. 1972, pp. 319–343.
  26. ZAREMBA, S. K.: Some applications of multidimensional integration by parts, Ann. Polon. Math. 21 (1968), 8–96.
DOI: https://doi.org/10.2478/udt-2023-0005 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X
Language: English
Page range: 65 - 82
Submitted on: Sep 16, 2022
Accepted on: Feb 27, 2023
Published on: Aug 10, 2023
Published by: Slovak Academy of Sciences, Mathematical Institute
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year

© 2023 Nathan Kirk, Florian Pausinger, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.