The existence of efficient lattice rules for multidimensional numerical integration
HTML articles powered by AMS MathViewer
- by Harald Niederreiter PDF
- Math. Comp. 58 (1992), 305-314 Request permission
Abstract:
In this contribution to the theory of lattice rules for multidimensional numerical integration, we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant. Then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants, which extends an earlier result of the author for lattice rules of rank 1.References
- Seymour Haber, Parameters for integrating periodic functions of several variables, Math. Comp. 41 (1983), no. 163, 115–129. MR 701628, DOI 10.1090/S0025-5718-1983-0701628-X G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon Press, Oxford, 1960.
- Edmund Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140–151 (German). MR 143329, DOI 10.1007/BF01387711
- Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192
- N. M. Korobov, Approximate evaluation of repeated integrals, Dokl. Akad. Nauk SSSR 124 (1959), 1207–1210 (Russian). MR 0104086
- L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
- Gerhard Larcher, A best lower bound for good lattice points, Monatsh. Math. 104 (1987), no. 1, 45–51. MR 903774, DOI 10.1007/BF01540524
- J. N. Lyness, Some comments on quadrature rule construction criteria, Numerical integration, III (Oberwolfach, 1987) Internat. Schriftenreihe Numer. Math., vol. 85, Birkhäuser, Basel, 1988, pp. 117–129. MR 1021529, DOI 10.1007/978-3-0348-6398-8_{1}2
- J. N. Lyness, An introduction to lattice rules and their generator matrices, IMA J. Numer. Anal. 9 (1989), no. 3, 405–419. MR 1011399, DOI 10.1093/imanum/9.3.405
- J. N. Lyness and I. H. Sloan, Some properties of rank-$2$ lattice rules, Math. Comp. 53 (1989), no. 188, 627–637. MR 982369, DOI 10.1090/S0025-5718-1989-0982369-6
- Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI 10.1090/S0002-9904-1978-14532-7
- Harald Niederreiter, Existence of good lattice points in the sense of Hlawka, Monatsh. Math. 86 (1978/79), no. 3, 203–219. MR 517026, DOI 10.1007/BF01659720
- Harald Niederreiter, Quasi-Monte Carlo methods for multidimensional numerical integration, Numerical integration, III (Oberwolfach, 1987) Internat. Schriftenreihe Numer. Math., vol. 85, Birkhäuser, Basel, 1988, pp. 157–171. MR 1021532, DOI 10.1007/978-3-0348-6398-8_{1}5
- Harald Niederreiter and Ian H. Sloan, Lattice rules for multiple integration and discrepancy, Math. Comp. 54 (1990), no. 189, 303–312. MR 995212, DOI 10.1090/S0025-5718-1990-0995212-4
- Ian H. Sloan, Lattice methods for multiple integration, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 131–143. MR 793949, DOI 10.1016/0377-0427(85)90012-3
- Ian H. Sloan and P. Kachoyan, Lattices for multiple integration, Mathematical programming and numerical analysis workshop (Canberra, 1983) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 6, Austral. Nat. Univ., Canberra, 1984, pp. 147–165. MR 794170
- Ian H. Sloan and Philip J. Kachoyan, Lattice methods for multiple integration: theory, error analysis and examples, SIAM J. Numer. Anal. 24 (1987), no. 1, 116–128. MR 874739, DOI 10.1137/0724010
- Ian H. Sloan and James N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp. 52 (1989), no. 185, 81–94. MR 947468, DOI 10.1090/S0025-5718-1989-0947468-3
- I. H. Sloan and J. N. Lyness, Lattice rules: projection regularity and unique representations, Math. Comp. 54 (1990), no. 190, 649–660. MR 1011443, DOI 10.1090/S0025-5718-1990-1011443-1
- Ian H. Sloan and Linda Walsh, A computer search of rank-$2$ lattice rules for multidimensional quadrature, Math. Comp. 54 (1990), no. 189, 281–302. MR 1001485, DOI 10.1090/S0025-5718-1990-1001485-4
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 58 (1992), 305-314
- MSC: Primary 65D30; Secondary 11K45
- DOI: https://doi.org/10.1090/S0025-5718-1992-1106976-5
- MathSciNet review: 1106976