Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature
SIAM Journal on Scientific Computing
Global Stochastic Optimization with Low-Dispersion Point Sets
Operations Research
The existence of good extensible rank-1 lattices
Journal of Complexity
Variance Reduction via Lattice Rules
Management Science
Finite-order weights imply tractability of multivariate integration
Journal of Complexity
On Korobov Lattice Rules in Weighted Spaces
SIAM Journal on Numerical Analysis
Constructing Embedded Lattice Rules for Multivariate Integration
SIAM Journal on Scientific Computing
Weighted compound integration rules with higher order convergence for all N
Numerical Algorithms
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Extensible lattice sequences have been proposed and studied in [F.J. Hickernell, H.S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, in: G.H. Golub, S.H. Lui, F.T. Luk, R.J. Plemmons (Eds.), Proceedings of the Workshop on Scientific Computing (Hong Kong), Singapore, Springer, Berlin, 1997, pp. 209-215; F.J. Hickernell, H.S. Hong, P. L'Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117-1138; F.J. Hickernell, H.Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286-300]. For the special case of extensible Korobov sequences, parameters can be found in [F.J. Hickernell, H.S. Hong, P. L'Ecuyer, C.Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117-1138]. The searches made to obtain these parameters were based on quality measures that look at several projections of the lattice. Because it is often the case in practice that low-dimensional projections are very important, it is of interest to find parameters for these sequences based on measures that look more closely at these projections. In this paper, we prove the existence of ''good'' extensible Korobov rules with respect to a quality measure that considers two-dimensional projections. We also report results of experiments made on different problems where the newly obtained parameters compare favorably with those given in [F.J. Hickernell, H.S. Hong, P. L'Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117-1138].