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The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration Cover

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

By: Takashi Goda  
Open Access
|Jul 2017

References

  1. [1] BILYK, D.: On Roth’s orthogonal function method in discrepancy theory, Uniform Distrib. Theory 6 (2011), 143-184.
  2. [2] DAVENPORT, H.: Note on irregularities of distribution, Mathematika 3 (1956), 131-135.10.1112/S0025579300001807
  3. [3] DICK, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal. 46 (2008), 1519-1553.10.1137/060666639
  4. [4] DICK, J.: Discrepancy bounds for infinite-dimensional order two digital sequences over F2, J. Number Theory 136 (2014), 204-232.10.1016/j.jnt.2013.09.012
  5. [5] DICK, J.-NUYENS, D.-PILLICHSHAMMER, F.: Lattice rules for nonperiodic smooth integrands, Numer. Math. 126 (2014), 259-291.10.1007/s00211-013-0566-0
  6. [6] DICK, J.-PILLICHSHAMMER, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511761188
  7. [7] GODA, T.: On the Lp discrepancy of two-dimensional folded Hammersley point sets, Arch. Math. 103 (2014), 389-398.10.1007/s00013-014-0698-1
  8. [8] GODA, T.-SUZUKI, K.-YOSHIKI,T.: The b-adic tent transformation for quasi-Monte Carlo integration using digital nets, J. Approx. Theory 194 (2015), 62-86.10.1016/j.jat.2015.02.002
  9. [9] GODA, T.-SUZUKI, K.-YOSHIKI, T.: Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration, J. Complexity (2016) 33 (2016), 30-54.10.1016/j.jco.2015.09.005
  10. [10] HALÁSZ, G.: On Roth’s method in the theory of irregularities of point distributions, in:Recent Progress in Analytic Number Theory, Academic Press, London, 1981, pp. 79-94.
  11. [11] HINRICHS, A.-KRITZINGER, R.-PILLICHSHAMMER, F.: Optimal order of Lp-discrepancy of digit shifted Hammersley point sets in dimension 2, Unif. Distrib. Theory 10 (2015), 115-133.
  12. [12] KRITZINGER, R.: Lp- and Sr p,qB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases, J. Complexity 33 (2016), 145-168.10.1016/j.jco.2015.10.002
  13. [13] LARCHER, G.-PILLICHSHAMMER, F.: Walsh series analysis of the L2 discrepancy of symmetrisized point sets, Monatsh. Math. 132 (2001), 1-18.10.1007/s006050170054
  14. [14] LARCHER, G.-PILLICHSHAMMER, F.: On the L2 discrepancy of the Sobol- Hammersley net in dimension 3, J. Complexity 18 (2002), 415-448.10.1006/jcom.2001.0606
  15. [15] MARKHASIN, L.: Lp- and Sr p,qB-discrepancy of (order 2) digital nets, Acta Arith. 168 (2015), 139-159.10.4064/aa168-2-4
  16. [16] MATOUŠEK, J.: Geometric Discrepancy. Springer-Verlag, Berlin, 1999.10.1007/978-3-642-03942-3
  17. [17] NIEDERREITER, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 63. SIAM, Philadelphia, 1992.10.1137/1.9781611970081
  18. [18] PILLICHSHAMMER, F.: On the Lp-discrepancy of the Hammersley point set, Monatsh. Math. 136 (2002), 7-79.10.1007/s006050200034
  19. [19] PONTRYAGIN, L. S.: Topological Groups. Gordon and Breach Science Publishers, Inc., New York, 1966.
  20. [20] PROINOV, P. D.: Symmetrization of the van der Corput generalized sequences, Proc. Japan Acad. Ser. A 64 (1988), 159-162.10.2183/pjab.64.159
  21. [21] ROTH, K. F.: On irregularities of distribution, Mathematika 1 (1954), 73-79.10.1112/S0025579300000541
  22. [22] SCHMIDT, W. M.: Irregularities of distribution, VI, Acta Arith. 21 (1972), 45-50.10.4064/aa-21-1-45-50
  23. [23] SCHMIDT, W. M.: Irregularities of distribution, X, in: Number Theory and Algebra, Academic Press, New York, 1977, pp. 311-329.
  24. [24] SLOAN, I. H.-JOE, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford, 1994.
  25. [25] SLOAN, I. H.-WO´ZNIAKOWSKI H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14 (1998), 1-33.
DOI: https://doi.org/10.1515/udt-2017-0001 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X
Language: English
Page range: 1 - 25
Submitted on: Nov 4, 2015
Accepted on: Nov 30, 2015
Published on: Jul 22, 2017
Published by: Slovak Academy of Sciences, Mathematical Institute
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year

© 2017 Takashi Goda, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.