Abstract
In this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube [0,1]s from a weighted reproducing kernel Hilbert space. We assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications: (i) It is usually not known in practice how to choose the weights. Thus by assuming that the weights are random variables, we obtain robust constructions (with respect to the weights) of lattice rules. This, to some extend, removes the necessity to carefully choose the weights. (ii) In practice it is convenient to use the same lattice rule for many different integrands. The best choice of weights for each integrand may vary to some degree, hence considering the weights random variables does justice to how lattice rules are used in applications. In this paper the worst-case error is therefore a random variable depending on random weights. We show how one can construct lattice rules which perform well for weights taken from a set with large measure. Such lattice rules are therefore robust with respect to certain changes in the weights. The construction algorithm uses the component-by-component (cbc) idea based on two criteria, one using the mean of the worst case error and the second criterion using a bound on the variance of the worst-case error. We call the new algorithm the cbc2c (component-by-component with 2 constraints) algorithm. We also study a generalized version which uses r constraints which we call the cbcrc (component-by-component with r constraints) algorithm. We show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights γ (1), . . . , γ (r). Thus, in applications, instead of finding one set of weights, it is enough to find a convex polytope in which the optimal weights lie. The price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule. Thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules. Numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights.
Similar content being viewed by others
References
Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
Cools R., Kuo F.Y., Nuyens D.: Constructing embedded lattice rules for multivariable integration. SIAM J. Sci. Comput. 28, 2162–2188 (2006)
Dick J.: On the convergence rate of the component-by-component construction of good lattice rules. J. Complexity 20, 493–522 (2004)
Dick J., Pillichshammer F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)
Dick J., Pillichshammer F., Waterhouse B.: The construction of good extensible rank-1 lattices. Math. Comput. 77, 2345–2373 (2008)
Hickernell, F.J.: Lattice rules: how well do they measure up? Random and quasi-random point sets, pp. 109–166. Lecture Notes in Statist., vol. 138. Springer, New York (1998)
Hickernell F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67, 299–322 (1998)
Hickernell F.J., Woźniakowski H.: Integration and approximation in arbitrary dimensions. High dimensional integration. Adv. Comput. Math. 12, 25–58 (2000)
Korobov N.M.: Approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959)
Korobov N.M.: Teoretiko-chislovye metody v priblizhennom analize. Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1963)
Kuo F.Y.: Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. Numerical integration and its complexity (Oberwolfach, 2001). J. Complexity 19, 301–320 (2003)
Larcher G., Leobacher G., Scheicher K.: On the tractability of the Brownian bridge algorithm. J. Complexity 19, 511–528 (2003)
Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)
Novak E., Woźniakowski H.: Tractability of Multivariate Problems. Linear Information, vol. 1. EMS Tracts in Mathematics, 6. European Mathematical Society (EMS), Zürich (2008)
Novak E., Woźniakowski H.: Tractability of Multivariate Problems. Standard Information for Functionals, vol. II. EMS Tracts in Mathematics, 12. European Mathematical Society (EMS), Zürich (2010)
Nuyens D., Cools R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75, 903–920 (2006)
Nuyens D., Cools R.: Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complexity 22, 4–28 (2006)
Nuyens, D., Cools, R.: Fast component-by-component construction, a reprise for different kernels. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 373–387. Springer, Berlin (2006)
Rosser J.B., Schoenfeld L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)
Sinescu V., L’Ecuyer P.: Existence and construction of shifted lattice rules with an arbitrary number of points and bounded weighted star discrepancy for general decreasing weights. J. Complexity 27, 449–465 (2011)
Sinescu, V., L’Ecuyer, P.: Variance bounds and existence results for randomly shifted lattice rules. J. Comput. Appl. Math. (2012, to appear)
Sloan I.H.: Finite-order integration weights can be dangerous. Comput. Methods Appl. Math. 7, 239–254 (2007)
Sloan I.H., Joe S.: Lattice Methods for Multiple Integration. Oxford Science Publications/The Clarendon Press/Oxford University Press, New York (1994)
Sloan I.H., Kuo F.Y., Joe S.: On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput. 71, 1609–1640 (2002)
Sloan I.H., Kuo F.Y., Joe S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal 40, 1650–1665 (2002)
Sloan I.H., Reztsov A.V.: Component-by-component construction of good lattice rules. Math. Comput. 71, 263–273 (2002)
Sloan I.H., Woźniakowski H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?. J. Complexity 14, 1–33 (1998)
Wang X.: Constructing robust good lattice rules for computational finance. SIAM J. Sci. Comput. 29, 598–621 (2007)
Wang X., Sloan I.H.: Efficient weighted lattice rules with applications to finance. SIAM J. Sci. Comput. 28, 728–750 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Dick was supported by a Queen Elizabeth 2 Fellowship from the Australian Research Council.
Rights and permissions
About this article
Cite this article
Dick, J. Random weights, robust lattice rules and the geometry of the cbcrc algorithm. Numer. Math. 122, 443–467 (2012). https://doi.org/10.1007/s00211-012-0469-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-012-0469-5