Abstract
We propose an alternative to the algorithm from Cools, Kuo, Nuyens (Computing 87(1–2):63–89, 2010), for constructing lattice rules with good trigonometric degree. The original algorithm has construction cost \(O(\vert {\mathcal{A}}_{d}(m)\vert + dN\log N)\) for an N-point lattice rule in d dimensions having trigonometric degree m, where the set \({\mathcal{A}}_{d}(m)\) has exponential size in both d and m (in the “unweighted degree” case, which is what we consider here). We reduce the cost to \(O(dN{(\log N)}^{2})\) with an implicit constant governing the needed precision (which is dependent on N and d).
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References
M. Beckers and R. Cools. A relation between cubature formulae of trigonometric degree and lattice rules. In H. Brass and G. Hämmerlin, editors, Numerical integration IV (Oberwolfach, 1992), pages 13–24. Birkhäuser Verlag, 1993.
R. Cools. More about cubature formulas and densest lattice packings. East Journal on Approximations, 12(1):37–42, 2006.
R. Cools and H. Govaert. Five- and six-dimensional lattice rules generated by structured matrices. J. Complexity, 19(6):715–729, 2003.
R. Cools, F. Y. Kuo, and D. Nuyens. Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput., 28(6):2162–2188, 2006.
R. Cools, F. Y. Kuo, and D. Nuyens. Constructing lattice rules based on weighted degree of exactness and worst case error. Computing, 87(1–2):63–89, 2010.
R. Cools and J. N. Lyness. Three- and four-dimensional K-optimal lattice rules of moderate trigonometric degree. Math. Comp., 70(236):1549–1567, 2001.
R. Cools, E. Novak, and K. Ritter. Smolyak’s construction of cubature formulas of arbitrary trigonometric degree. Computing, 62(2):147–162, 1999.
R. Cools and D. Nuyens. A Belgian view on lattice rules. In A. Keller, S. Heinrich, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2006, pages 3–21. Springer-Verlag, 2008.
R. Cools and D. Nuyens. Extensions of Fibonacci lattice rules. In P. L’Écuyer and A. B. Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008, pages 1–12. Springer-Verlag, 2009.
R. Cools and A. V. Reztsov. Different quality indexes for lattice rules. J. Complexity, 13(2):235–258, 1997.
R. Cools and I. H. Sloan. Minimal cubature formulae of trigonometric degree. Math. Comp., 65(216):1583–1600, 1996.
J. A. De Loera, J. Rambau, and F. Santos. Triangulations, volume 25 of Algorithms and Computation in Mathematics. Springer-Verlag, 2010.
J. Dick, F. Pillichshammer, G. Larcher, and H. Woźniakowski. Exponential convergence and tractability of multivariate integration for Korobov spaces. Math. Comp., 80(274):905–930, 2011.
I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press, 7th edition, 2007.
F. J. Hickernell. Lattice rules: How well do they measure up? In P. Hellekalek and G. Larcher, editors, Random and Quasi-Random Point Sets, pages 109–166. Springer-Verlag, Berlin, 1998.
J. N. Lyness. Notes on lattice rules. J. Complexity, 19(3):321–331, 2003.
J. N. Lyness and T. Sørevik. Four-dimensional lattice rules generated by skew-circulant matrices. Math. Comp., 73(245):279–295, 2004.
J. N. Lyness and T. Sørevik. Five-dimensional K-optimal lattice rules. Math. Comp., 75(255):1467–1480, 2006.
H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. Number 63 in Regional Conference Series in Applied Mathematics. SIAM, 1992.
D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp., 75(254):903–920, 2006.
D. Nuyens and R. Cools. Fast component-by-component construction, a reprise for different kernels. In H. Niederreiter and D. Talay, editors, Monte Carlo and Quasi-Monte Carlo Methods 2004, pages 371–385. Springer-Verlag, 2006.
D. Nuyens and R. Cools. Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complexity, 22(1):4–28, 2006.
N. N. Osipov, R. Cools, and M. V. Noskov. Extremal lattices and the construction of lattice rules. Appl. Math. Comput., 217(9):4397–4407, 2011.
I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration. Oxford Science Publications, 1994.
I. H. Sloan and A. V. Reztsov. Component-by-component construction of good lattice rules. Math. Comp., 71(237):263–273, 2002.
I. H. Sloan and H. Woźniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complexity, 14(1):1–33, 1998.
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The authors would like to thank the two anonymous referees for useful comments on the manuscript.
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Achtsis, N., Nuyens, D. (2012). A Component-by-Component Construction for the Trigonometric Degree. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_10
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