Numerische Mathematik

, Volume 134, Issue 1, pp 163–196 | Cite as

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

  • Aicke Hinrichs
  • Lev Markhasin
  • Jens Oettershagen
  • Tino Ullrich


We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces \(S^r_{p,q}B(\mathbb {T}^d)\) with dominating mixed smoothness \(1/p<r<2\). We show that order 2 digital nets achieve the optimal rate of convergence \(N^{-r} (\log N)^{(d-1)(1-1/q)}\). The logarithmic term does not depend on r and hence improves the known bound of Dick (SIAM J Numer Anal 45:2141–2176, 2007) for the special case of Sobolev spaces \(H^r_{\text {mix}}(\mathbb {T}^d)\). Secondly, the rate of convergence is independent of the integrability p of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of order 2 digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case \(r=2\).

Mathematics Subject Classification

11K06 11J71 42C10 46E35 65C05 65D30 65D32 91G60 


  1. 1.
    Amanov, T.I.: Spaces of Differentiable Functions with Dominating Mixed Derivatives. Nauka Kaz SSR, Alma-Ata (1976)Google Scholar
  2. 2.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bakhvalov, N.S.: Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions. Zh. Vychisl. Mat. Mat. Fiz. 4(4), 5–63 (1963)MathSciNetGoogle Scholar
  4. 4.
    Bilyk, D.: On Roth’s orthogonal function method in discrepancy theory. Unif. Distrib. Theory 6, 143–184 (2011)MathSciNetMATHGoogle Scholar
  5. 5.
    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 1–123 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dick, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dick, J.: Discrepancy bounds for infinite-dimensional order two digital sequences over \(\mathbb{F}_2\). J. Number Theory 136, 204–232 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dick, J., Kritzer, P.: Duality theory and propagation rules for generalized digital nets. Math. Comput. 79, 993–1017 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dick, J., Kuo, F., Thong Le Gia, Q., Schwab, C.: Multi-level higher order QMC Galerkin discretization for affine parametric operator equations (2014) (arXiv e-prints)Google Scholar
  11. 11.
    Dick, J., Le Gia, Q.T., Schwab, C.: Higher order quasi-monte carlo integration for holomorphic, parametric operator equations (2014) (arXiv e-prints)Google Scholar
  12. 12.
    Dick, J., Niederreiter, H.: On the exact \(t\)-value of Niederreiter and Sobol sequences. J. Complex. 24(56), 572–581 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  14. 14.
    Dick, J., Pillichshammer, F.: Discrepancy theory and quasi-Monte Carlo integration. In: Panorama in Discrepancy Theory. Springer, Berlin (2013)Google Scholar
  15. 15.
    Dick, J., Pillichshammer, F.: Explicit constructions of point sets and sequences with low discrepancy. In: Uniform Distribution and Quasi-Monte Carlo Methods—Discrepancy, Integration and Applications, pp. 63–86 (2014)Google Scholar
  16. 16.
    Dick, J., Pillichshammer, F.: Optimal \(L_2\)-discrepancy bounds for higher order digital sequences over the finite field \(\mathbb{F}_2\). Acta Arith. 162(1), 65–99 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dubinin, V.V.: Cubature formulas for classes of functions with bounded mixed difference. Mat. USSR Sb. 76, 283–292 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dubinin, V.V.: Cubature formulae for Besov classes. Iz. Math. 61(2), 259–283 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dũng, D.: B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness. J. Complex. 27(6), 541–567 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Dũng, D., Ullrich, T.: Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square. Math. Nachr. doi:10.1002/mana.201400048
  21. 21.
    Faber, G.: Über stetige Funktionen. Math. Ann. 66, 81–94 (1909)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Glasserman, P.: Monte Carlo Methods in Financial Engineering. Applications of Mathematics: Stochastic Modelling and Applied Probability. Springer, Berlin (2004)Google Scholar
  23. 23.
    Halton, J.H.: Algorithm 247: radical-inverse quasi-random point sequence. Commun. ACM 7(12), 701–702 (1964)CrossRefGoogle Scholar
  24. 24.
    Hinrichs, A.: Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness. Math. Nachr. 283(3), 478–488 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hinrichs, A.: Discrepancy, integration and tractability. In: Dick, J., Kuo, F.Y., Peters, G.W., Sloan, I.H. (eds.) Monte Carlo and Quasi-Monte Carlo. Methods, vol. 2012 (2014)Google Scholar
  26. 26.
    Hinrichs, A., Oettershagen, J.: Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives. In: Proceedings of the Conference MCQMC, Leuven (2014) (to appear)Google Scholar
  27. 27.
    Hlawka, E.: Zur angenäherten Berechnung mehrfacher Integrale. Monatsh. Math. 66, 140–151 (1962)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Korobov, N.M.: Approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959)MathSciNetMATHGoogle Scholar
  29. 29.
    Kuipers, H.N.L.: Uniform Distribution of Sequences. Wiley, New York (1974)MATHGoogle Scholar
  30. 30.
    Markhasin, L.: Discrepancy and integration in function spaces with dominating mixed smoothness. Dissert. Math. 494, 1–81 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Markhasin, L.: Discrepancy of generalized Hammersley type point sets in Besov spaces with dominating mixed smoothness. Unif. Distrib. Theory 8(1), 135–164 (2013)MathSciNetMATHGoogle Scholar
  32. 32.
    Markhasin, L.: Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension. J. Complex. 29(5), 370–388 (2013)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Markhasin, L.: \({L}_2\)- and \({S}^r_{p, q}{B}\)-discrepancy of (order 2) digital nets. Acta Arith. 168(2), 139–160 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Matoušek, J.: Geometric Discrepancy. An Illustrated Guide. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
  35. 35.
    Nguyen, V.K., Ullrich, M., Ullrich, T.: Boundedness of pointwise multiplication and change of variable and applications to numerical integration (2015) (preprint)Google Scholar
  36. 36.
    Niederreiter, H.: Point sets and sequences with small discrepancy. Monatsh. Math. 104, 273–337 (1987)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Niederreiter, H., Xing, C.: A construction of low-discrepancy sequences using global function fields. Acta Arith. 73(1), 87–102 (1995)MathSciNetMATHGoogle Scholar
  38. 38.
    Niederreiter, H., Xing, C.: Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl. 2(3), 241–273 (1996)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Nikol’skij, S.M.: Approximation of Functions of Several Variables and Embedding Theorems. Nauka, Moskva (1977)Google Scholar
  40. 40.
    Novak, E.: Some results on the complexity of numerical integration. In: Monte Carlo and Quasi-Monte Carlo Methods. Proceedings of the Conference MCQMC, Leuven (2014) (to appear)Google Scholar
  41. 41.
    Novak, E., Woźniakowski, H.: Tractability of multivariate problems. Standard information for functionals, vol. II. In: EMS Tracts in Mathematics, vol. 12. European Mathematical Society (EMS), Zürich (2010)Google Scholar
  42. 42.
    Pirsic, G.: A software implementation of Niederreiter–Xing sequences. In: Monte Carlo and Quasi-Monte Carlo Methods, vol. 2000. Springer, Berlin (2002)Google Scholar
  43. 43.
    Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. A Wiley-Interscience Publication. Wiley, Chichester (1987)Google Scholar
  44. 44.
    Skriganov, M.M.: Constructions of uniform distributions in terms of geometry of numbers. J. Complex. 6, 200–230 (1994)MathSciNetMATHGoogle Scholar
  45. 45.
    Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963)MATHGoogle Scholar
  46. 46.
    Temlyakov, V.N.: On reconstruction of multivariate periodic functions based on their values at the knots of number-theoretical nets. Anal. Math. 12, 287–305 (1986)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Temlyakov, V.N.: On a way of obtaining lower estimates for the errors of quadrature formulas. Mat. Sb. 181(10), 1403–1413 (1990)Google Scholar
  48. 48.
    Temlyakov, V.N.: Error estimates for Fibonacci quadrature formulas for classes of functions with a bounded mixed derivative. Trudy Mat. Inst. Steklov 200, 327–335 (1991)MathSciNetMATHGoogle Scholar
  49. 49.
    Temlyakov, V.N.: Approximation of Periodic Functions. Computational Mathematics and Analysis Series. Nova Science, Commack (1993)MATHGoogle Scholar
  50. 50.
    Temlyakov, V.N.: Cubature formulas, discrepancy, and nonlinear approximation. J. Complex. 19(3), 352–391 (2003) (numerical integration and its complexity, Oberwolfach, 2001)Google Scholar
  51. 51.
    Triebel, H.: Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration. In: EMS Tracts in Mathematics, vol. 11. European Mathematical Society (EMS), Zürich (2010)Google Scholar
  52. 52.
    Triebel, H.: EMS Series of Lectures in Mathematics. Faber systems and their use in sampling, discrepancy, numerical integration. European Mathematical Society (EMS), Zürich (2012)Google Scholar
  53. 53.
    Ullrich, M., Ullrich, T.: The role of Frolov’s cubature formula for functions with bounded mixed derivative (2015). arXiv:1503.08846 [math.NA] (arXiv e-prints)
  54. 54.
    Ullrich, T.: Function spaces with dominating mixed smoothness; characterization by differences. Jenaer Schriften Math. Inf. (2006) (Math/Inf/05/06)Google Scholar
  55. 55.
    Ullrich, T.: Smolyak’s algorithm, sampling on sparse grids and Sobolev spaces of dominating mixed smoothness. East J. Approx. 14(1), 1–38 (2008)MathSciNetMATHGoogle Scholar
  56. 56.
    Ullrich, T.: Optimal cubature in Besov spaces with dominating mixed smoothness on the unit square. J. Complex. 30, 72–94 (2014)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Wahba, G.: Smoothing noisy data with spline functions. Numer. Math. 24(5), 383–393 (1975)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Aicke Hinrichs
    • 1
  • Lev Markhasin
    • 2
  • Jens Oettershagen
    • 3
  • Tino Ullrich
    • 3
  1. 1.Institute for AnalysisJohannes Kepler UniversityLinzAustria
  2. 2.Institute for Stochastics and ApplicationsUniversity of StuttgartStuttgartGermany
  3. 3.Institute for Numerical SimulationUniversity of BonnBonnGermany

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