Abstract
In this paper we consider numerical integration of smooth functions lying in a particular reproducing kernel Hilbert space. We show that the worst-case error of numerical integration in this space converges at the optimal rate, up to some power of a log N factor. A similar result is shown for the mean square worst-case error, where the bound for the latter is always better than the bound for the square worst-case error. Finally, bounds for integration errors of functions lying in the reproducing kernel Hilbert space are given. The paper concludes by illustrating the theory with numerical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1971)
Caflisch, R.E., Morokoff, W.J., Owen, A.B.: Valuation of mortgage backed securities using Brownian Bridge to reduce effective dimension. J. Comput. Finance 1, 27–46 (1997)
Chrestenson, H.E.: A class of generalized Walsh functions. Pac. J. Math. 5, 17–31 (1955)
Dick, J.: Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45, 2141–2176 (2007)
Dick, J.: The decay of the Walsh coefficients of smooth functions. Bull. Aust. Math. Soc. (2009, to appear)
Dick, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008)
Dick, J., Baldeaux, J.: Equidistribution properties of generalized nets and sequences. In: L’Ecuyer, P., Owen. A.B. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer (2010, to appear)
Dick, J., Kritzer, P.: Duality theory and propagation rules for digital nets of higher order. Math. Comput. (2010, to appear)
Dick, J., Niederreiter, H.: On the exact t-value of Niederreiter and Sobol’ sequences. J. Complex. 24, 572–581 (2008)
Dick, J., Pillichshammer, F.: Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complex. 21, 149–195 (2005)
Dick, J., Pillichshammer, F.: On the mean square weighted ℒ2 discrepancy of randomized digital (t,m,s)-nets over ℤ2. Acta Arith. 117, 371–403 (2005)
Dick, J., Kuo, F., Pillichshammer, F., Sloan, I.H.: Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comput. 74, 1895–1921 (2005)
Dick, J., Kritzer, P., Pillichshammer, F., Schmid, W.Ch.: On the existence of higher order polynomial lattices based on a generalized figure of merit. J. Complex. 23, 581–593 (2007)
Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41, 337–351 (1982)
Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18, 209–232 (1998)
L’Ecuyer, P., Lemieux, C.: Variance reduction via lattice rules. Manag. Sci. 46, 1214–1235 (2000)
L’Ecuyer, P., Lemieux, C.: Recent advances in randomized quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovszki, F. (eds.) Modeling Uncertainty. Internat. Ser. Oper. Res. Management Sci., vol. 46, pp. 419–474. Kluwer Academic, Boston (2002)
Matoušek, J.: Geometric Discrepancy. Algorithms Combin., vol. 18. Springer, Berlin (1999)
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)
Niederreiter, H., Pirsic, G.: Duality for digital nets and its applications. Acta Arith. 97, 173–182 (2001)
Niederreiter, H., Xing, Ch.: Global function fields with many rational places and their applications. In: Mullin, R.C., Mullen, G.L. (eds.) Finite Fields: Theory, Applications, and Algorithms, Waterloo, ON, 1997. Contemp. Math., vol. 225, pp. 87–111. Amer. Math. Soc., Providence (1999)
Pirsic, G.: A software implementation of Niederreiter–Xing sequences. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, 2000 (Hong Kong), pp. 434–445. Springer, Berlin (2002)
Sharygin, I.F.: A lower estimate for the error of quadrature formulas for certain classes of functions. Zh. Vychisl. Mat. i Mat. Fiz. 3, 370–376 (1963)
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., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complex. 14, 1–33 (1998)
Sloan, I.H., Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Complex. 17, 697–721 (2001)
Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963)
Sobol’, I.M.: Distribution of points in a cube and approximate evaluation of integrals. Zh. Vychisl. Mat. i Mat. Fiz. 7, 784–802 (1967)
Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45, 5–24 (1923)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ian Sloan.
Rights and permissions
About this article
Cite this article
Baldeaux, J., Dick, J. QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach. Constr Approx 30, 495–527 (2009). https://doi.org/10.1007/s00365-009-9074-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-009-9074-y