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12 - Construction of a rank-1 lattice sequence based on primitive polynomials

Published online by Cambridge University Press:  18 December 2014

Alexander Keller
Affiliation:
NVIDIA, Berlin
Nikolaus Binder
Affiliation:
NVIDIA, Berlin
Carsten Wächter
Affiliation:
NVIDIA, Berlin
Gerhard Larcher
Affiliation:
Johannes Kepler Universität Linz
Friedrich Pillichshammer
Affiliation:
Johannes Kepler Universität Linz
Arne Winterhof
Affiliation:
Austrian Academy of Sciences, Linz
Chaoping Xing
Affiliation:
Nanyang Technological University, Singapore
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Publisher: Cambridge University Press
Print publication year: 2014

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References

[1] R., Cools, F., Kuo and D., Nuyens, Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput. 28, 2162–2188, 2006.Google Scholar
[2] R., Cranley and T., Patterson, Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13, 904–914, 1976.Google Scholar
[3] J., Dick, F., Pillichshammer and B., Waterhouse, The construction of good extensible rank-1 lattices. Math. Comp. 77, 2345–2373, 2008.Google Scholar
[4] I., Friedel and A., Keller, Fast generation of randomized low-discrepancy point sets. In: H., Niederreiter, K., Fang and F., Hickernell (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 257–273. Springer, 2002.
[5] H., Gill and C., Lemieux, Searching for extensible Korobov rules. J. Complexity 23, 603–613, 2007.Google Scholar
[6] L., Grünschloß, M., Raab and A., Keller, Enumerating quasi-Monte Carlo point sequences in elementary intervals. In: L., Plaskota and H., Woźniakowski (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 399–408. Springer, 2012.
[7] F., Hickernell and H., Hong, Computing multivariate normal probabilities using rank-1 lattice sequences. In: G., Golub, S., Lui, F., Luk and R., Plemmons (eds.), Proceedings of the Workshop on Scientific Computing in Hong Kong, pp. 209–215. Springer Verlag, Singapore, 1997.
[8] F., Hickernell and H., Niederreiter, The existence of good extensible rank-1 lattices. J. Complexity 19, 286–300, 2003.Google Scholar
[9] F., Hickernell, H., Hong, P., L'Ecuyer and C., Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J. Sci. Comput. 22, 1117–1138, 2001.Google Scholar
[10] S., Joe and F., Kuo, Constructing Sobol' sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30(5), 2635–2654, 2008.Google Scholar
[11] A., Keller, Quasi-Monte Carlo image synthesis in a nutshell. In: J., Dick, F., Kuo, G., Peters and I., Sloan (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2012, pp. 203–238. Springer, Heidelberg, 2013.
[12] A., Keller and L., Grünschloß, Parallel quasi-Monte Carlo integration by partitioning low discrepancy sequences. In: L., Plaskota and H., Woźniakowski (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 487–498. Springer, 2012.
[13] T., Kollig and A., Keller, Efficient multidimensional sampling (Proc. Eurographics 2002). Comput. Graphics Forum 21(3), 557–563, 2002.Google Scholar
[14] P., L'Ecuyer and D., Munger, Constructing adapted lattice rules using problem-dependent criteria. Proceedings of the 2012 Winter Simulation Conference, pp. 373–384. IEEE Press, New York, 2012.
[15] P., L'Ecuyer and D., Munger, Latticebuilder: a general software tool for constructing rank-1 lattice rules. ACM Trans. Math. Software, submitted 2012.Google Scholar
[16] R., Lidl and H., Niederreiter, Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge, 1986.
[17] E., Maize, Contributions to the theory of error reduction in quasi-Monte Carlo methods. PhD Thesis, Claremont Graduate School, 1980.
[18] E., Maize, J., Sepikas and J., Spanier, Accelerating the convergence of lattice methods by importance sampling-based transformations. In: L., Plaskota and H., Woźniakowski (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 557–572. Springer, 2012.
[19] H., Niederreiter, Quasirandom sampling in computer graphics. Proceedings 3rd International Seminar on Digital Image Processing in Medicine, Remote Sensing and Visualization of Information (Riga, Latvia), pp. 29–34, 1992.Google Scholar
[20] H., Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, PA, 1992.
[21] H., Niederreiter and F., Pillichshammer, Construction algorithms for good extensible lattice rules. Construct. Approx 30, 361–393, 2009.Google Scholar
[22] A., Owen, Monte Carlo extension of quasi-Monte Carlo. Proceedings of the 1998 Winter Simulation Conference, pp. 571–577. IEEE Press, New York, 1998.
[23] M., Saito and M., Matsumoto, SIMD-oriented fast Mersenne twister: a 128-bit pseudorandom number generator. In: A., Keller, S., Heinrich and H., Niederreiter (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 607–622. Springer, 2007.
[24] N., Saxena and E., McCluskey, Primitive polynomial generation algorithms implementation and performance analysis. Technical Report, Stanford University, Center for Reliable Computing (CRC), TR 04-03, 2004.
[25] I., Sloan and S., Joe, Lattice Methods for Multiple Integration. Clarendon Press, Oxford, 1994.
[26] I., Sobol', On the Distribution of points in a cube and the approximate evaluation of integrals. Zh. Vychisl. Mat. Mat. Fiz. 7(4), 784–802, 1967.Google Scholar

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