# Random weights, robust lattice rules and the geometry of the cbc*r*c algorithm

## 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 cbc*r*c (component-by-component with *r* constraints) algorithm. We show that lattice rules generated by the cbc*r*c 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.

### Mathematics Subject Classification

65D30 65D32## Preview

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### References

- 1.Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc.
**68**, 337–404 (1950)MathSciNetMATHCrossRefGoogle Scholar - 2.Cools R., Kuo F.Y., Nuyens D.: Constructing embedded lattice rules for multivariable integration. SIAM J. Sci. Comput.
**28**, 2162–2188 (2006)MathSciNetMATHCrossRefGoogle Scholar - 3.Dick J.: On the convergence rate of the component-by-component construction of good lattice rules. J. Complexity
**20**, 493–522 (2004)MathSciNetMATHCrossRefGoogle Scholar - 4.Dick J., Pillichshammer F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
- 5.Dick J., Pillichshammer F., Waterhouse B.: The construction of good extensible rank-1 lattices. Math. Comput.
**77**, 2345–2373 (2008)MathSciNetMATHCrossRefGoogle Scholar - 6.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)Google Scholar
- 7.Hickernell F.J.: A generalized discrepancy and quadrature error bound. Math. Comput.
**67**, 299–322 (1998)MathSciNetMATHCrossRefGoogle Scholar - 8.Hickernell F.J., Woźniakowski H.: Integration and approximation in arbitrary dimensions. High dimensional integration. Adv. Comput. Math.
**12**, 25–58 (2000)MathSciNetMATHCrossRefGoogle Scholar - 9.Korobov N.M.: Approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR
**124**, 1207–1210 (1959)MathSciNetMATHGoogle Scholar - 10.Korobov N.M.: Teoretiko-chislovye metody v priblizhennom analize. Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1963)MATHGoogle Scholar
- 11.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)MathSciNetMATHCrossRefGoogle Scholar - 12.Larcher G., Leobacher G., Scheicher K.: On the tractability of the Brownian bridge algorithm. J. Complexity
**19**, 511–528 (2003)MathSciNetMATHCrossRefGoogle Scholar - 13.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)Google Scholar
- 14.Novak E., Woźniakowski H.: Tractability of Multivariate Problems. Linear Information, vol. 1. EMS Tracts in Mathematics, 6. European Mathematical Society (EMS), Zürich (2008)CrossRefGoogle Scholar
- 15.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)CrossRefGoogle Scholar
- 16.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)MathSciNetMATHCrossRefGoogle Scholar - 17.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)MathSciNetMATHCrossRefGoogle Scholar - 18.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)Google Scholar
- 19.Rosser J.B., Schoenfeld L.: Approximate formulas for some functions of prime numbers. Ill. J. Math.
**6**, 64–94 (1962)MathSciNetMATHGoogle Scholar - 20.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)MathSciNetMATHCrossRefGoogle Scholar - 21.Sinescu, V., L’Ecuyer, P.: Variance bounds and existence results for randomly shifted lattice rules. J. Comput. Appl. Math. (2012, to appear)Google Scholar
- 22.Sloan I.H.: Finite-order integration weights can be dangerous. Comput. Methods Appl. Math.
**7**, 239–254 (2007)MathSciNetMATHGoogle Scholar - 23.Sloan I.H., Joe S.: Lattice Methods for Multiple Integration. Oxford Science Publications/The Clarendon Press/Oxford University Press, New York (1994)MATHGoogle Scholar
- 24.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)MathSciNetMATHCrossRefGoogle Scholar - 25.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)MathSciNetMATHCrossRefGoogle Scholar - 26.Sloan I.H., Reztsov A.V.: Component-by-component construction of good lattice rules. Math. Comput.
**71**, 263–273 (2002)MathSciNetMATHGoogle Scholar - 27.Sloan I.H., Woźniakowski H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?. J. Complexity
**14**, 1–33 (1998)MathSciNetMATHCrossRefGoogle Scholar - 28.Wang X.: Constructing robust good lattice rules for computational finance. SIAM J. Sci. Comput.
**29**, 598–621 (2007)MathSciNetMATHCrossRefGoogle Scholar - 29.Wang X., Sloan I.H.: Efficient weighted lattice rules with applications to finance. SIAM J. Sci. Comput.
**28**, 728–750 (2006)MathSciNetMATHCrossRefGoogle Scholar