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Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

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Abstract

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.

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References

  1. Baldeaux, J., Dick, J.: QMC rules of arbitrary high order: reproducing kernel Hilbert space approach. Constr. Approx. 30(3), 495–527 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baldeaux, J., Dick, J., Greslehner, J., Pillichshammer, F.: Construction algorithms for higher order polynomial lattice rules. J. Complex. 27(3–4), 281–299 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dick, J.: Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Stat. 39(3), 1372–1398 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dick, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46(3), 1519–1553 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dick, J.: The decay of the Walsh coefficients of smooth functions. Bull. Aust. Math. Soc. 80, 430–453 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dick, J., Kritzer, P., Pillichshammer, F., Schmid, W.C.: On the existence of higher order polynomial lattices based on a generalized figure of merit. J. Complex. 23(4–6), 581–593 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dick, J., Kuo, F.Y., Pillichshammer, F., Sloan, I.H.: Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comput. 74(252), 1895–1921 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dick, J., Pillichshammer, F.: Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complex. 21(2), 149–195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dick, J., Pillichshammer, F.: Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules. J. Complex. 23(4–6), 436–453 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  12. Korobov, N.M.: The approximate computation of multiple integrals/approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959) in Russian

    MathSciNet  MATH  Google Scholar 

  13. Korobov, N.M.: Number-Theoretic Methods in Approximate Analysis. Goz. Izdat. Fiz.-Math (1963) in Russian

  14. Larcher, G., Lauss, A., Niederreiter, H., Schmid, W.C.: Optimal polynomials for (t, m, s)-nets and numerical integration of multivariate Walsh series. SIAM J. Numer. Anal. 33(6), 2239–2253 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, no. 63. Regional Conference Series in Applied Mathematics. SIAM (1992)

  16. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems—vol. I: Linear Information. EMS Tracts in Mathematics, vol. 6. European Mathematical Society Publishing House (2008)

  17. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems—vol. II: Standard Information for Functionals. EMS Tracts in Mathematics, vol. 12. European Mathematical Society Publishing House (2010)

  18. Nussbaumer, H.J.: Fast Fourier Transform and Convolution Algorithms, 2nd edn. Springer-Verlag (1982)

  19. Nuyens, D.: Fast construction of good lattice rules. Ph.D. Thesis, Dept. of Computer Science, K. U. Leuven (2007)

  20. Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75(254), 903–920 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nuyens, D., Cools, R.: Fast component-by-component construction, a reprise for different kernels. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 371–385. Springer-Verlag (2006)

  22. Owen, A.B.: Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34(5), 1884–1910 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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, pp. 434–445. Springer-Verlag (2002)

  24. Schmid, W.C.: Improvements and extensions of the “Salzburg Tables” by using irreducible polynomials. In: Niederreiter, H., Spanier, J. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 436–447. Springer, Berlin (2000)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  26. Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex. 14(1), 1–33 (1998)

    Article  MATH  Google Scholar 

  27. Stahnke, W.: Primitive binary polynomials. Math. Comput. 27(124), 977–980 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Šarygin, I.F.: Lower bounds for the error of quadrature formulas on classes of functions. USSR Comput. Math. Math. Phys. 3, 489–497 (1965); Translation from Russian Zh. Vychisl. Mat. Mat. Fiz. 3, 370–376 (1963)

    Google Scholar 

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Authors and Affiliations

  1. School of Finance and Economics, The University of Technology, Sydney, NSW, 2007, Australia

    Jan Baldeaux

  2. School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW, 2052, Australia

    Josef Dick

  3. Institut für Finanzmathematik, Universität Linz, Altenbergerstraße 69, 4040, Linz, Austria

    Gunther Leobacher & Friedrich Pillichshammer

  4. Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A – bus 2402, 3001, Heverlee, Belgium

    Dirk Nuyens

Authors
  1. Jan Baldeaux
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  2. Josef Dick
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  3. Gunther Leobacher
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  4. Dirk Nuyens
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  5. Friedrich Pillichshammer
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Corresponding author

Correspondence to Friedrich Pillichshammer.

Additional information

Josef Dick is supported by a Queen Elizabeth 2 Fellowship from the Australian Research Council.

Gunther Leobacher is partially supported by the Austrian Science Foundation (FWF), Project P21196.

Dirk Nuyens is a postdoctoral fellow of the Research Foundation Flanders (FWO).

Friedrich Pillichshammer is partially supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

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Baldeaux, J., Dick, J., Leobacher, G. et al. Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules. Numer Algor 59, 403–431 (2012). https://doi.org/10.1007/s11075-011-9497-y

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  • DOI: https://doi.org/10.1007/s11075-011-9497-y

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