In Search for Good Chebyshev Lattices

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

Recently we introduced a new framework to describe some point sets used for multivariate integration and approximation (Cools and Poppe, BIT Numer Math 51:275–288, 2011), which we called Chebyshev lattices. The associated integration rules are equal weight rules, with corrections for the points on the boundary. In this text we detail the development of exhaustive search algorithms for good Chebyshev lattices where the cost of the rules, i.e., the number of points needed for a certain degree of exactness, is used as criterium. Almost loopless algorithms are considered to avoid dependencies on the rank of the Chebyshev lattice and the dimension. Also, several optimisations are applied: reduce the vast search space by exploiting symmetries, lower the cost of the point set creation and minimise the cost of the degree verification. The concluding summary of the search results indicates that higher rank rules in general are better and that the blending formulae due to Godzina lead to the best rules within the class of Chebyshev lattice rules: no better rules have been found in the searches conducted in up to five dimensions.

Notes

Acknowledgements

This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State Science Policy Office. The scientific responsibility rests with its authors.

The authors acknowledge the support for this project by the Bijzonder Onderzoeksfonds of the Katholieke Universiteit Leuven.

References

  1. 1.
    Caliari, M., De Marchi, S., Vianello, M.: Bivariate polynomial interpolation on the square at new nodal sets. Applied Mathematics and Computation 165(2), 261–274 (2005)Google Scholar
  2. 2.
    Cools, R., Nuyens, D.: A Belgian view on lattice rules. In: A. Keller, S. Heinrich, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 3–21. Springer (2008)Google Scholar
  3. 3.
    Cools, R., Poppe, K.: Chebyshev lattices, a unifying framework for cubature with the Chebyshev weight function. BIT Numerical Mathematics 51, 275–288 (2011)Google Scholar
  4. 4.
    Ehrlich, G.: Loopless algorithms for generating permutations, combinations, and other combinatorial configurations. Journal of the ACM 20(3), 500–513 (1973)Google Scholar
  5. 5.
    Godzina, G.: Dreidimensionale Kubaturformeln für zentralsymmetrische Integrale. Ph.D. thesis, Universität Erlangen-Nürnberg (1994)Google Scholar
  6. 6.
    Knuth, D.E.: Combinatorial Algorithms, The Art of Computer Programming, vol. 4 (2005–2009)Google Scholar
  7. 7.
    Lyness, J.N.: An introduction to lattice rules and their generator matrices. IMA Journal of Numerical Analysis 9, 405–419 (1989)Google Scholar
  8. 8.
    Möller, H.: Lower bounds for the number of nodes in cubature formulae. Numerische Integration 45, 221–230 (1979)Google Scholar
  9. 9.
    Morrow, C.R., Patterson, T.N.L.: Construction of algebraic cubature rules using polynomial ideal theory. SIAM Journal on Numerical Analysis 15(5), 953–976 (1978)Google Scholar
  10. 10.
    Noskov, M.: Analogs of Morrow-Patterson type cubature formulas. Journal of Computation Mathematics and Mathematical Physics 30, 1254–1257 (1991)Google Scholar
  11. 11.
    Poppe, K., Cools, R.: CHEBINT: Operations on multivariate Chebyshev approximations. https://lirias.kuleuven.be/handle/123456789/325973
  12. 12.
    Sloan, I.H.: Polynomial interpolation and hyperinterpolation over general regions. Journal of Approximation Theory 83(2), 238–254 (1995)Google Scholar
  13. 13.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford Science Publications (1994)Google Scholar
  14. 14.
    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Review 50(1), 67–87 (2008)Google Scholar
  15. 15.
    Trefethen, L.N., Hale, N., Platte, R.B., Driscoll, T.A., Pachón, R.: Chebfun version 3. Oxford University (2009). http://www.maths.ox.ac.uk/chebfun/

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenHeverleeBelgium

Personalised recommendations