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A Belgian View on Lattice Rules

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Monte Carlo and Quasi-Monte Carlo Methods 2006

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References

  1. N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337-404, 1950.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Beckers and R. Cools. A relation between cubature formulae of trigonometric degree and lattice rules. In H. Brass and G. Hämmerlin, editors, Numerical Integration IV, pages 13-24, Basel, 1993. Birkhäuser Verlag.

    Google Scholar 

  3. M. Bourdeau and A. Pitre. Tables of good lattices in four and five dimensions. Numer. Math., 47:39-43, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Cools. Constructing cubature formulae: the science behind the art, volume 6 of Acta Numerica, pages 1-54. Cambridge University Press, 1997.

    Google Scholar 

  5. R. Cools. More about cubature formulas and densest lattice packings. East Journal on Approximations, 12(1):37-42, 2006.

    MathSciNet  Google Scholar 

  6. R. Cools and H. Govaert. Five- and six-dimensional lattice rules gener-ated by structured matrices. J. Complexity, 19(6):715-729, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Cools, F. Y. Kuo and D. Nuyens. Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput., 28(6):2162-2188,2006.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Cools and J. Lyness. Three- and four-dimensional K -optimal lattice rules of moderate trigonometric degree. Math. Comp., 70(236):1549-1567,2001.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Cools, I. Mysovskikh and H. Schmid. Cubature Formulae and Orthogonal Polynomials. J. Comput. Appl. Math., 127:121-152, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Cools and D. Nuyens. The role of structured matrices for the construction of integration lattices. JNAIAM J. Numer. Anal. Ind. Appl. Math., 1(3):257-272, 2006.

    MATH  MathSciNet  Google Scholar 

  11. R. Cranley and T. Patterson. Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal., 13:904-914, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Cools and I. H. Sloan. Minimal cubature formulae of trigonometric degree. Math. Comp., 65(216):1583-1600, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  13. H.  Dammertz, A. Keller and S. Dammertz. Simulation on rank-1 lattices. In this volume, pages 205-212.

    Google Scholar 

  14. S.  Dammertz and A. Keller. Image synthesis by rank-1 lattices. In this volume, pages 217-236.

    Google Scholar 

  15. J. Dick and F. Y. Kuo. Constructing good lattice rules with millions of points. In Niederreiter [Nie04], pages 181-197.

    Google Scholar 

  16. J. Dick and F. Y. Kuo. Reducing the construction cost of the component-by-component construction of good lattice rules. Math. Comp., 73(248):1967-1988, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Dick, F. Pillichshammer and B. Waterhouse. The construction of good extensible rank-1 lattices. Math. Comp. To appear.

    Google Scholar 

  18. K. Frolov. On the connection between quadrature formulas and sub-lattices of the lattice of integral vectors. Dokl. Akad. Nauk SSSR, 232:40-43, 1977. (Russian) Soviet Math. Dokl. 18: 37-41, 1977 (En-glish).

    Google Scholar 

  19. P. Gruber and C. Lekkerkerker. Geometry of numbers. North Holland, 1987.

    Google Scholar 

  20. F. J. Hickernell. Lattice rules: How well do they measure up? In P. Hellekalek and G. Larcher, editors, Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 109-166. Springer-Verlag, 1998.

    Google Scholar 

  21. F. Hickernell. A generalized discrepancy and quadrature error bound. Math. Comp., 67(221):299-322, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. J. Hickernell. Obtaining O(n−2+ε ) convergence for lattice quadrature rules. In K. T. Fang, F. J. Hickernell and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2000, pages 274-289. Springer-Verlag, 2002.

    Google Scholar 

  23. F. J. Hickernell, H. S. Hong, P. L’Écuyer and C. Lemieux. Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J. Sci. Comput., 22:1117-1138, 2001.

    Article  Google Scholar 

  24. F. J. Hickernell and H. Niederreiter. The existence of good extensible rank-1 lattices. J. Complexity, 19(3):286-300, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Hill and I. Robinson. d2lri: A nonadaptive algorithm for two-dimensional cubature. J. Comput. Appl. Math., 112(1-2):121-145, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Hill and I. Robinson. Quadrature using 64-bit IEEE arithmetic for integrands over [0, 1] with a singularity at 1. Theoret. Comput. Sci., 351(1):82-100, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Joe. Component by component construction of rank-1 lattice rules having O(n−1 (ln(n))d ) star discrepancy. In Niederreiter [Nie04], pages 293-298.

    Google Scholar 

  28. S. Joe and I. H. Sloan. Embedded lattice rules for multidimensional integration. SIAM J. Numer. Anal., 29:1119-1154, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Joe and V. Sinescu. Good lattice rules based on the general weighted star discrepancy. Math. Comp., 76(258):989-1004, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  30. A. Keller. Stratification by rank-1  lattices. In Niederreiter  [Nie04], pages 299-313.

    Google Scholar 

  31. N. Korobov. On approximate calculation of multiple integrals. Dokl. Akad. Nauk SSSR, 124:1207-1210, 1959. (Russian).

    MATH  MathSciNet  Google Scholar 

  32. N. Korobov. Properties and calculation of optimal coefficients. Dokl. Akad. Nauk SSSR, 132:1009-1012, 1960. (Russian) Soviet Math. Dokl. 1: 696-700, 1960 (English).

    Google Scholar 

  33. F. Y. Kuo. Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. J. Complexity, 19:301-320, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  34. F. Y. Kuo, W. T. M. Dunsmuir, I. H. Sloan, M. P. Wand and R. S. Womersley. Quasi-Monte Carlo for highly structured generalised response models. Methodology and Computing in Applied Probability. To appear.

    Google Scholar 

  35. P. L’ Écuyer and C. Lemieux. Recent advances in randomized quasi-Monte Carlo methods. In M. Dror, P. L’Ecuyer and F. Szidarovszki, editors, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pages 419-474. Kluwer Academic Publishers, 2002.

    Google Scholar 

  36. J. Lyness. An introduction to lattice rules and their generator matrices. IMA J. Numer. Anal., 9:405-419, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Lyness. Notes on lattice rules. J. Complexity, 19(3):321-331, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Lyness and T. Sørevik. Four-dimensional lattice rules generated by skew-circulant matrices. Math. Comp., 73(245):279-295, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  39. J. Lyness and T. Sørevik. Five-dimensional k-optimal lattice rules. Math. Comp., 75(255): 1467-1480, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  40. D. Maisonneuve. Recherche et utilisation des “bons treillis”.Programmation et résultats numerériques. In S. Zaremba, editor, Ap-plications of Number Theory to Numerical Analysis, pages 121-201. Academic Press, 1972.

    Google Scholar 

  41. H. Minkowski. Gesammelte Abhandlungen. Chelsea Publishing Com-pany, New York, Reprinted (originally published, in 2 volumes, Leipzig, 1911) edition, 1967.

    Google Scholar 

  42. I. Mysovskikh. Interpolatory Cubature Formulas. Izdat. ‘Nauka’, Moscow-Leningrad, 1981. (Russian).

    Google Scholar 

  43. I. Mysovskikh. Quadrature formulae of the highest trigonometric degree of accuracy. Zh. vychisl. Mat. mat. Fiz., 25:1246-1252, 1985. (Russian) U.S.S.R. Comput. Maths. Math. Phys. 25:180-184, 1985 (English).

    Google Scholar 

  44. I. Mysovskikh. On cubature formulas that are exact for trigonometric polynomials. Dokl. Akad. Nauk SSSR, 296:28-31, 1987. (Russian) Soviet Math. Dokl. 36:229-232, 1988 (English).

    Google Scholar 

  45. I. Mysovskikh. On the construction of cubature formulas that are exact for trigonometric polynomials. In A. Wakulicz, editor, Numerical Analysis and Mathematical Modelling, volume 24 of Banach Center Publications, pages 29-38. PWN - Polish Scientific Publishers, Warsaw, 1990.(Russian).

    Google Scholar 

  46. D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp., 75(2):903-920, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  47. D. Nuyens and R. Cools. Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complexity, 22(1):4-28, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  48. H. Niederreiter. Quasi-Monte Carlo methods and pseudo-random num-bers. Bull. Amer. Math. Soc., 84(6):957-1041, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  49. H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of CBMS-NSF regional conference series in applied mathematics. SIAM, Philadelphia, 1992

    MATH  Google Scholar 

  50. H. Niederreiter, editor. Monte-Carlo and Quasi-Monte Carlo Methods - 2002. Springer-Verlag, 2004.

    Google Scholar 

  51. M. Noskov. Cubature formulae for the approximate integration of periodic functions. Metody Vychisl., 14:15-23, 1985. (Russian).

    MATH  MathSciNet  Google Scholar 

  52. M. Noskov. Formulas for the approximate integration of periodic func-tions. Metody Vychisl., 15:19-22, 1988. (Russian).

    MATH  MathSciNet  Google Scholar 

  53. N. N. Osipov. Cubature formulas for periodic functions. Ph.D. thesis, Krasnoyarsk State Technical University, 2004. (Russian).

    Google Scholar 

  54. N. N. Osipov and A. V. Petrov. Construction of sequences of lattice rules which are exact for trigonometric polynomials in four variables. Vychisl. Tekhnol., 9, Spec. Iss. 1:102-110, 2004. (Russian).

    MATH  Google Scholar 

  55. J. Radon. Zur mechanischen Kubatur. Monatsh. Math., 52:286-300, 1948.

    Article  MATH  MathSciNet  Google Scholar 

  56. I. Robinson and E. deDoncker. Algorithm 45: Automatic computation of improper integrals over a bounded or unbounded planar region. Computing, 27:253-284, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  57. M. Revers. Numerical integration of the Radon transform on classes \(E_s^\alpha\) in multiple finite dimensions. Computing, 54(2):147-165, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  58. A. Reztsov. On cubature formulas of Gaussian type with an asymptotic minimal number of nodes. Mathematicheskie Zametki, 48:151-152, 1990.

    MATH  MathSciNet  Google Scholar 

  59. I. Robinson and M. Hill. Algorithm 816: r2d2lri: an algorithm for automatic two-dimensional cubature. ACM Trans. Math. Software, 28 (1):75-100, 2002.

    Article  MATH  Google Scholar 

  60. A. Semenova. Computing experiments for construction of cubature formulae of high trigonometric accuracy. In M. Ramazanov, editor, Cubature Formulas and their Applications (Russian), pages 105-115, Ufa, 1996.

    Google Scholar 

  61. A. Semenova. An algorithm for the construction of cubature formulas of high trigonometric accuracy. In C. Shoynjurov, editor, Cubature Formulas and their Applications (Russian), pages 93-105, Ulan-Ude, 1997.

    Google Scholar 

  62. V. Sinescu and S. Joe. Good lattice rules with a composite number of points based on the product weighted star discrepancy. In this volume, pages 645-658.

    Google Scholar 

  63. I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration. Oxford University Press, 1994.

    Google Scholar 

  64. I. H. Sloan and P. Kachoyan. Lattice mathods for multiple integration: theory, error analysis and examples. SIAM J. Numer. Anal., 24:116-128, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  65. I. H. Sloan, F. Y. Kuo and S. Joe. Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal., 40(5):1650-1665,2002.

    Article  MATH  MathSciNet  Google Scholar 

  66. I. H. Sloan, F. Y. Kuo and S. Joe. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comp., 71(240):1609-1640,2002.

    Article  MATH  MathSciNet  Google Scholar 

  67. I. H. Sloan and A. V. Reztsov. Component-by-component construction of good lattice rules. Math. Comp., 71(237):263-273, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  68. A. Stroud. Quadrature methods for functions of more than one variable. New York Acad. Sci., 86:776-791, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  69. N. Temirgaliev. Application of divisor theory to the numerical integra-tion of periodic functions of several variables. Math. USSR Sbornik, 69 (2):527-542, 1991.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Computer Science, K.U.Leuven, Celestijnenlaan 200A, B-3001, Heverlee, Belgium

    Ronald Cools & Dirk Nuyens

Authors
  1. Ronald Cools
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  2. Dirk Nuyens
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Editors and Affiliations

  1. Department of Computer Science, Ulm University, Albert-Einstein-Allee 11, 89069, Ulm, Germany

    Alexander Keller

  2. Department of Computer Science, University of Kaiserslautern, 67653, Kaiserslautern, Germany

    Stefan Heinrich

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

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Cools, R., Nuyens, D. (2008). A Belgian View on Lattice Rules. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_1

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