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Construction of scrambled polynomial lattice rules over \(\mathbb{F}_{2}\) with small mean square weighted \(\mathcal{L}_{2}\) discrepancy

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Abstract

The \(\mathcal{L}_{2}\) discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Woźniakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted \(\mathcal{L}_{2}\) discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field \(\mathbb{F}_{2}\) whose scrambled point sets have small mean square weighted \(\mathcal{L}_{2}\) discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of N −2+δ for all δ>0, where N denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol’ sequences.

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References

  1. Baldeaux, J.: Higher order nets and sequences. Ph.D. thesis, The University of New South Wales (2010)

  2. Baldeaux, J., Dick, J.: A construction of polynomial lattice rules with small gain coefficients. Numer. Math. 119, 271–297 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Caflisch, R.E., Morokoff, W., Owen, A.B.: Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Comput. Finance 1, 27–46 (1997)

    Google Scholar 

  4. Chrestenson, H.E.: A class of generalized Walsh functions. Pac. J. Math. 5, 17–31 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cranley, R., Patterson, T.N.L.: Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13, 904–914 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. Dick, J., Pillichshammer, F.: Optimal \(\mathcal{L}_{2}\) discrepancy bounds for higher order digital sequences over finite field \(\mathbb{F}_{2}\). Preprint. Available at arXiv:1207.5189

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Dick, J., Leobacher, G., Pillichshammer, F.: Construction algorithms for digital nets with low weighted star discrepancy. SIAM J. Numer. Anal. 43, 76–95 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fine, N.J.: On the Walsh functions. Trans. Am. Math. Soc. 65, 372–414 (1949)

    Article  MATH  Google Scholar 

  12. Hickernell, F.J.: The mean square discrepancy of randomized nets. ACM Trans. Model. Comput. Simul. 6, 274–296 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67, 299–322 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Joe, S., Kuo, F.Y.: Constructing Sobol’ sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30, 2635–2654 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Kritzer, P., Pillichshammer, F.: Constructions of general polynomial lattices for multivariate integration. Bull. Aust. Math. Soc. 76, 93–110 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kritzer, P., Pillichshammer, F.: On the component by component construction of polynomial lattice point sets for numerical integration in weighted Sobolev spaces. Unif. Dist. Theory 6, 79–100 (2011)

    MATH  MathSciNet  Google Scholar 

  18. L’Ecuyer, P.: Polynomial integration lattices. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 73–98. Springer, Berlin (2004)

    Chapter  Google Scholar 

  19. Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer Series in Statistics. Springer, New York (2009)

    MATH  Google Scholar 

  20. Matoušek, J.: On the \(\mathcal{L}_{2}\)-discrepancy for anchored boxes. J. Complex. 14, 527–556 (1998)

    Article  MATH  Google Scholar 

  21. Matoušek, J.: Geometric Discrepancy: An Illustrated Guide, Algorithms and Combinatorics. Springer, Berlin (1999)

    Book  Google Scholar 

  22. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  23. Niederreiter, H.: Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslov. Math. J. 42, 143–166 (1992)

    MATH  MathSciNet  Google Scholar 

  24. Ninomiya, S., Tezuka, S.: Toward real-time pricing of complex financial derivatives. Appl. Math. Finance 3, 1–20 (1996)

    Article  MATH  Google Scholar 

  25. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume II: Standard Informations for Functionals. EMS, Zurich (2010)

    Book  Google Scholar 

  26. 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, 903–920 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. 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. 373–387. Springer, Berlin (2006)

    Chapter  Google Scholar 

  28. Owen, A.B.: Randomly permuted (t,m,s)-nets and (t,s)-sequences. In: Niederreiter, H., Shiue, J.-S. (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, pp. 299–317. Springer, New York (1995)

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Owen, A.B.: Scrambled net variance for integrals of smooth functions. Ann. Stat. 25, 1541–1562 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Paskov, S.H., Traub, J.F.: Faster valuation of financial derivatives. J. Portf. Manag. 22, 113–120 (1995)

    Article  Google Scholar 

  32. Pillichshammer, F.: Polynomial lattice point sets. In: Plaskota, L., Woźniakowski, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 189–210. Springer, Berlin (2012)

    Chapter  Google Scholar 

  33. Roth, K.F.: On irregularities of distribution. Mathematika 1, 73–79 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  37. Sobol’, I.M.: Distribution of points in a cube and approximate evaluation of integrals. Zh. Vychisl. Mat. Mat. Fiz. 7, 784–802 (1967). (in Russian)

    MathSciNet  Google Scholar 

  38. Tezuka, S., Faure, H.: I-binomial scrambling of digital nets and sequences. J. Complex. 19, 744–757 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45, 5–24 (1923)

    Article  MATH  Google Scholar 

  40. Woźniakowski, H.: Average case complexity of multivariate integration. Bull., New Ser., Am. Math. Soc. 24, 185–194 (1991)

    Article  MATH  Google Scholar 

  41. Zaremba, S.K.: Some applications of multidimensional integration by parts. Ann. Pol. Math. 21, 85–96 (1968)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-8656, Tokyo

    Takashi Goda

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  1. Takashi Goda
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Correspondence to Takashi Goda.

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Communicated by Lothar Reichel.

The support of Grant-in-Aid for JSPS Fellows No. 24-4020 is gratefully acknowledged.

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Goda, T. Construction of scrambled polynomial lattice rules over \(\mathbb{F}_{2}\) with small mean square weighted \(\mathcal{L}_{2}\) discrepancy. Bit Numer Math 54, 401–423 (2014). https://doi.org/10.1007/s10543-013-0459-8

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