On the step-by-step construction of quasi–Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces
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- by I. H. Sloan, F. Y. Kuo and S. Joe PDF
- Math. Comp. 71 (2002), 1609-1640 Request permission
Abstract:
We develop and justify an algorithm for the construction of quasi–Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found “component-by-component”: the ($d+1$)-th component of the generator vector and the shift are obtained by successive $1$-dimensional searches, with the previous $d$ components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for $n$-point rules with $n$ prime and all dimensions 1 to $d$ requires a total cost of $O(n^3d^2)$ operations. This may be reduced to $O(n^3d)$ operations at the expense of $O(n^2)$ storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and $n$ up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.References
- Fred J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp. 67 (1998), no. 221, 299–322. MR 1433265, DOI 10.1090/S0025-5718-98-00894-1
- Fred J. Hickernell, Lattice rules: how well do they measure up?, Random and quasi-random point sets, Lect. Notes Stat., vol. 138, Springer, New York, 1998, pp. 109–166. MR 1662841, DOI 10.1007/978-1-4612-1702-2_{3}
- O. P. Iliev, M. S. Kaschiev, S. D. Margenov, Bl. H. Sendov, and P. S. Vassilevski (eds.), Recent advances in numerical methods and applications. II, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1786606, DOI 10.1142/4061
- Novak, E. and Woźniakowski, H. (2001), Intractability results for integration and discrepancy, J. Complexity, 17, 388–441.
- Harald Niederreiter, 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, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
- I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1442955
- Sloan, I. H. and Reztsov, A. V. (2002). Component-by-component construction of good lattice rules, Math. Comp., 17, 263–273.
- Danfu Han and Xinghua Wang, Convergence on a deformed Newton method, Appl. Math. Comput. 94 (1998), no. 1, 65–72. MR 1627389, DOI 10.1016/S0096-3003(97)10066-2
- Sloan, I. H. and Woźniakowski, H. (2001). Tractability of multivariate integration for weighted Korobov classes, J. Complexity, 17, 697–721.
- Woźniakowski, H. (1999), Efficiency of quasi-Monte Carlo algorithms for high-dimensional integrals, Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.), Springer-Verlag, Berlin, 114–136.
Additional Information
- I. H. Sloan
- Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
- MR Author ID: 163675
- ORCID: 0000-0003-3769-0538
- Email: sloan@maths.unsw.edu.au
- F. Y. Kuo
- Affiliation: Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand
- MR Author ID: 703418
- Email: f.kuo@math.waikato.ac.nz
- S. Joe
- Affiliation: Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand
- Email: stephenj@math.waikato.ac.nz
- Received by editor(s): October 30, 2000
- Published electronically: March 20, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Math. Comp. 71 (2002), 1609-1640
- MSC (2000): Primary 65D30, 65D32; Secondary 68Q25
- DOI: https://doi.org/10.1090/S0025-5718-02-01420-5
- MathSciNet review: 1933047