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Duality theory and propagation rules for generalized digital nets
Author(s):
Josef
Dick;
Peter
Kritzer.
Journal:
Math. Comp.
79
(2010),
993-1017.
MSC (2000):
Primary 11K38, 11K45, 65C05, 94B05
Posted:
November 17, 2009
MathSciNet review:
2600553
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Additional information
Abstract:
Digital nets are used in quasi-Monte Carlo algorithms for approximating high dimensional integrals over the unit cube. Hence one wants to have explicit constructions of digital nets of high quality. In this paper we consider the so-called propagation rules for digital nets, which state how one can obtain a new digital net of different size from existing digital nets. This way one often can generate digital nets of higher quality than were previously known. Here we generalize existing propagation rules for classical digital nets to generalized digital nets as introduced by Dick.
References:
-
- 1.
- Dick, J., Baldeaux, J.: Equidistribution properties of generalized nets and sequences. To apear in: L'Ecuyer, P. and Owen, A. (eds.): Monte Carlo and Quasi-Monte Carlo Methods 2008, 2010.
- 2.
- Bierbrauer, J., Edel, Y., Schmid, W.Ch.: Coding-theoretic constructions for
 -nets and ordered orthogonal arrays. J. Comb. Des. 10, 403-418, 2002. MR 1932120 (2003k:94047) - 3.
- Blackmore, N., Norton, G.H.: Matrix-product codes over
 . Applicable Algebra Engrg. Comm. Comput. 12, 477-500, 2001. MR 1873271 (2002m:94054) - 4.
- Dick, J.: Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45, 2141-2176, 2007. MR 2346374 (2008h:11076)
- 5.
- Dick, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519-1553, 2008. MR 2391005 (2009d:42077)
- 6.
- Dick, J., Kritzer, P., Pillichshammer, F., Schmid, W.Ch.: On the existence of higher order polynomial lattices based on a generalized figure of merit. J. Complexity 23, 581-593, 2007. MR 2372015 (2009a:65054)
- 7.
- Niederreiter, H.: Point sets and sequences with small discrepancy. Monatsh. Math. 104, 273-337, 1987. MR 918037 (89c:11120)
- 8.
- Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Series in Applied Mathematics 63, SIAM, Philadelphia, 1992. MR 1172997 (93h:65008)
- 9.
- Niederreiter, H.: Constructions of
 -nets. In: Niederreiter, H. and Spanier. J. (eds.): Monte Carlo and Quasi-Monte Carlo Methods 1998. Springer, Berlin, 2000, pp. 70-85. MR 1849843 (2002e:65012) - 10.
- Niederreiter, H.: Constructions of
 -nets and  -sequences. Finite Fields Appl. 11, 578-600, 2005. MR 2158777 (2006c:11090) - 11.
- Niederreiter, H., Özbudak, F.: Matrix-product constructions of digital nets. Finite Fields Appl. 10, 464-479, 2004. MR 2067609 (2005f:11289)
- 12.
- Niederreiter, H., Pirsic, G.: Duality for digital nets and its applications. Acta Arith. 97, 173-182, 2001. MR 1824983 (2001m:11130)
- 13.
- Niederreiter, H., Xing, C.P.: Nets,
 -sequences, and algebraic geometry. In: Hellekalek, P. and Larcher, G. (eds.): Random and Quasi-Random Point Sets. Springer, New York, 1998, pp. 267-302. MR 1662844 (99k:11121) - 14.
- Niederreiter, H., Xing, C.P.: Constructions of digital nets. Acta Arith. 102, 189-197, 2002. MR 1889629 (2003a:11092)
- 15.
- R. Schürer, W.Ch. Schmid: MinT--the database of optimal net, code, OA, and OOA parameters. Available at: http://mint.sbg.ac.at (November 17, 2009).
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Additional Information:
Josef
Dick
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia
Email:
josef.dick@unsw.edu.au
Peter
Kritzer
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia
Email:
peter.kritzer@gmail.com
DOI:
10.1090/S0025-5718-09-02315-1
PII:
S 0025-5718(09)02315-1
Keywords:
Quasi-Monte Carlo,
numerical integration,
digital nets,
duality theory,
propagation rules
Received by editor(s):
August 29, 2008
Received by editor(s) in revised form:
April 9. 2009
Posted:
November 17, 2009
Additional Notes:
The support of the Australian Research Council under its Centre of Excellence program is gratefully acknowledged.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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