# Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

- 218 Downloads
- 12 Citations

## Abstract

Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced by Dick (Ann Stat 39:1372–1398, 2011) and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the \(s\)-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is \(ds\), where the integer \(d\) is the so-called interlacing factor. In this paper we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen’s full scrambling scheme as well as the simplifications studied by Hickernell, Matoušek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to \(\alpha \ge 1\), and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base \(b\) and dimension \(ds\), to which we then apply the interlacing scheme of order \(d\), we obtain a construction cost of the algorithm of order \(\mathcal {O}(dsmb^m)\) operations using \(\mathcal {O}(b^m)\) memory in case of product weights, where \(b^m\) is the number of points in the polynomial lattice point set.

## Keywords

Randomized quasi-Monte Carlo Interlaced scrambled polynomial lattice rules Component-by-component construction Weighted function spaces Tractability of multivariate integration## Mathematics Subject Classification

Primary 65C05 Secondary 65D30, 65D32## Notes

### Acknowledgments

The first author is supported by JSPS Grant-in-Aid for JSPS Fellows No. 24-4020, and the second author is supported by a Queen Elizabeth 2 Fellowship (DP1097023) from the Australian Research Council. T.G. would like to thank Josef Dick for his hospitality while visiting the University of New South Wales where this research was carried out.

## References

- 1.J. Baldeaux, Higher order nets and sequences, Ph.D. thesis, The University of New South Wales, 2010.Google Scholar
- 2.J. Baldeaux and J. Dick, A construction of polynomial lattice rules with small gain coefficients. Numer. Math. 119 (2011), 271–297.Google Scholar
- 3.J. Baldeaux, J. Dick, J. Greslehner and F. Pillichshammer, Construction algorithms for higher order polynomial lattice rules. J. Complexity 27 (2011), 281–299.Google Scholar
- 4.J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens and F. Pillichshammer, Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules. Numer. Algorithms 59 (2012) 403–431.Google Scholar
- 5.H. E. Chrestenson, A class of generalized Walsh functions. Pacific J. Math. 5 (1955) 17–31.Google Scholar
- 6.J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45 (2007) 2141–2176.Google Scholar
- 7.J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46 (2008) 1519–1553.Google Scholar
- 8.J. Dick, On quasi-Monte Carlo rules achieving higher order convergence, Springer, Berlin, 2009, pp. 73–96.Google Scholar
- 9.J. Dick, Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39 (2011) 1372–1398.Google Scholar
- 10.J. Dick and M. Gnewuch, Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. J. Approx. Theory 184 (2014) 111–145.Google Scholar
- 11.J. Dick, F.Y. Kuo, F. Pillichshammer and I.H. Sloan, Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comp. 74 (2005) 1895–1921.Google Scholar
- 12.J. Dick, G. Leobacher, and F. Pillichshammer, Construction algorithms for digital nets with low weighted star discrepancy. SIAM. J. Numer. Anal. 43 (2005) 76–95.Google Scholar
- 13.J. Dick and F. Pillichshammer, Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules. J. Complexity 23 (2007) 436–453.Google Scholar
- 14.J. Dick and F. Pillichshammer,
*Digital nets and sequences: discrepancy theory and quasi-Monte Carlo integration*. Cambridge University Press, Cambridge, 2010.Google Scholar - 15.J. Dick, I. H. Sloan, X. Wang and H. Woźniakowski, Good lattice rules in weighted Korobov spaces with general weights. Numer. Math. 103 (2006) 63–97.Google Scholar
- 16.H. Faure, Discrépances de suites associées à un système de numération (en dimension s). Acta Arith. 41 (1982) 337–351.Google Scholar
- 17.F. J. Hickernell, The mean square discrepancy of randomized nets. ACM Trans. Modeling Comput. Simul. 6 (1996) 274–296.Google Scholar
- 18.N. M. Korobov, The approximate computation of multiple integrals/approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR 124 (1959) 1207–1210.Google Scholar
- 19.P. Kritzer and F. Pillichshammer, Constructions of general polynomial lattices for multivariate integration. Bull. Austral. Math. Soc. 76 (2007) 93–110.Google Scholar
- 20.L. Kuipers and H. Niederreiter,
*Uniform distribution of sequences*. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974.Google Scholar - 21.G. Larcher, A. Lauss, H. Niederreiter and W. Ch. Schmid, Optimal polynomials for \((t, m, s)\)-nets and numerical integration of multivariate Walsh series. SIAM. J. Numer. Anal. 33 (1996) 2239–2253.Google Scholar
- 22.C. Lemieux and P. L’Ecuyer, Randomized polynomial lattice rules for multivariate integration and simulation. SIAM. J. Sci. Comput. 24 (2003) 1768–1789.Google Scholar
- 23.J. Matoušek, On the \(L_2\) discrepancy for anchored boxes. J. Complexity 14 (1998) 527–556.Google Scholar
- 24.T. Müller-Gronbach, E. Novak and K. Ritter, Monte Carlo-Algorithmen. (German) Springer-Lehrbuch. Springer, Heidelberg, 2012.Google Scholar
- 25.H. Niederreiter, Low-discrepancy and low-dispersion sequences. J. Number Theory 30 (1988) 51–70.Google Scholar
- 26.H. Niederreiter,
*Random number generation and quasi-Monte Carlo methods*. in: CBMS-NSF Series in Applied Mathematics, vol. 63, SIAM, Philadelphia, 1992.Google Scholar - 27.H. Niederreiter, Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. J. 42 (1992) 143–166.Google Scholar
- 28.H. Niederreiter and C. P. Xing,
*Rational points on curves over finite fields: theory and applications*. London Mathematical Society Lecture Note Series, 285. Cambridge University Press, Cambridge, 2001.Google Scholar - 29.E. Novak,
*Deterministic and stochastic error bounds in numerical analysis*. Lecture Notes in Mathematics, 1349. Springer-Verlag, Berlin, 1988.Google Scholar - 30.E. Novak and H. Woźniakowski,
*Tractability of multivariate problems. Vol. 1: Linear information*. EMS Tracts in Mathematics, 6. European Mathematical Society (EMS), Zürich, 2008.Google Scholar - 31.E. Novak and H. Woźniakowski,
*Tractability of multivariate problems. Volume II: Standard information for functionals*. EMS Tracts in Mathematics, 12. European Mathematical Society (EMS), Zürich, 2010.Google Scholar - 32.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 (2006) 903–920.Google Scholar
- 33.D. Nuyens and R. Cools, Fast component-by-component construction, a reprise for different kernels. Springer, Berlin, 2006, pp. 373–387.Google Scholar
- 34.A. B. Owen, Randomly permuted \((t, m, s)\)-nets and \((t, s)\)-sequences. in Monte Carlo and quasi-Monte Carlo Methods in Scientific Computing, Springer, New York, 1995, pp. 299–317.Google Scholar
- 35.A. B. Owen, Monte Carlo variance of scrambled net quadrature. SIAM. J. Numer. Anal. 34 (1997) 1884–1910.Google Scholar
- 36.A. B. Owen, Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 (1997) 1541–1562.Google Scholar
- 37.A. B. Owen, Variance with alternative scramblings of digital nets. ACM Trans. Model. Comp. Simul. 13 (2003) 363–378.Google Scholar
- 38.I. H. Sloan and A. V. Reztsov, Component-by-component construction of good lattice rules. Math. Comp. 71 (2002) 263–273.Google Scholar
- 39.I. H. Sloan and H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14 (1998) 1–33.Google Scholar
- 40.I. M. Sobol’, The distribution of points in a cube and approximate evaluation of integrals. Zh. Vycisl. Mat. i Mat. Fiz. 7 (1967) 784–802.Google Scholar
- 41.J. L. Walsh, A closed set of normal orthogonal functions. Amer. J. Math. 45 (1923) 5–24.Google Scholar