The Analysis of Vertex Modified Lattice Rules in a Non-periodic Sobolev Space

  • Dirk Nuyens
  • Ronald Cools


In a series of papers, in 1993, 1994 and 1996, Ian Sloan together with Harald Niederreiter introduced a modification of lattice rules for non-periodic functions, called “vertex modified lattice rules”, and a particular breed called “optimal vertex modified lattice rules”, see Numerical Integration IV (Birkhäuser 1993) pp. 253–265, J Comput Appl Math 51(1):57–70, 1994, and Comput Math Model 23(8–9):69–77, 1996. These are like standard lattice rules but they distribute the point at the origin to all corners of the unit cube, either by equally distributing the weight and so obtaining a multi-variate variant of the trapezoidal rule, or by choosing weights such that multilinear functions are integrated exactly. In the 1994 paper, Niederreiter and Sloan concentrate explicitly on Fibonacci lattice rules, which are a particular good choice of 2-dimensional lattice rules. Error bounds in this series of papers were given related to the star discrepancy.

In this paper we pose the problem in terms of the so-called unanchored Sobolev space, which is a reproducing kernel Hilbert space often studied nowadays in which functions have L2-integrable mixed first derivatives. It is known constructively that randomly shifted lattice rules, as well as deterministic tent-transformed lattice rules and deterministic fully symmetrized lattice rules can achieve close to O(N−1) convergence in this space, see Sloan et al. (Math Comput 71(240):1609–1640, 2002) and Dick et al. (Numer Math 126(2):259–291, 2014) respectively, where possible \(\log ^{s}(N)\) terms are taken care of by weighted function spaces.

We derive a break down of the worst-case error of vertex modified lattice rules in the unanchored Sobolev space in terms of the worst-case error in a Korobov space, a multilinear space and some additional “mixture term”. For the 1-dimensional case this worst-case error is obvious and gives an explicit expression for the trapezoidal rule. In the 2-dimensional case this mixture term also takes on an explicit form for which we derive upper and lower bounds. For this case we prove that there exist lattice rules with a nice worst-case error bound with the additional mixture term of the form \(N^{-1}\log ^{2}(N)\).



We thank Jens Oettershagen for useful comments and pointers to [4, 18]. We also thank the Taiwanese National Center for Theoretical Sciences (NCTS)—Mathematics Division, and the National Taiwan University (NTU)—Department of Mathematics, where part of this work was carried out. We thank the referees for their helpful comments and acknowledge financial support from the KU Leuven research fund (OT:3E130287 and C3:3E150478).


  1. 1.
    Cools, R., Kuo, F.Y., Nuyens, D.: Constructing lattice rules based on weighted degree of exactness and worst case error. Computing 87(1–2), 63–89 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dick, J., Nuyens, D., Pillichshammer, F.: Lattice rules for nonperiodic smooth integrands. Numer. Math. 126(2), 259–291 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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. 288, 743–762 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67(221), 299–322 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. In: Regional Conference Series in Applied Mathematics, vol. 63, SIAM, Philadelphia (1992)Google Scholar
  7. 7.
    Niederreiter, H., Sloan, I.H.: Quasi-Monte Carlo methods with modified vertex weights. In: Brass, H., Hämmerlin, G. (eds.) Numerical Integration IV (Oberwolfach, 1992), International Series of Numerical Mathematics, pp. 253–265. Birkhäuser, Basel (1993)Google Scholar
  8. 8.
    Niederreiter, H., Sloan, I.H.: Integration of nonperiodic functions of two variables by Fibonacci lattice rules. J. Comput. Appl. Math. 51(1), 57–70 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Niederreiter, H., Sloan, I.H.: Variants of the Koksma–Hlawka inequality for vertex-modified Quasi-Monte Carlo integration rules. Comput. Math. Model. 23(8–9), 69–77 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems — Volume I: Linear Information. EMS Tracts in Mathematics, vol. 6. European Mathematical Society Publishing House, Zürich (2008)Google Scholar
  11. 11.
    Nuyens, D.: The construction of good lattice rules and polynomial lattice rules. In: Kritzer, P., Niederreiter, H., Pillichshammer, F., Winterhof, A. (eds.) Radon Series on Computational and Applied Mathematics, pp. 223–256. De Gruyter, Berlin (2014)Google Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Reddy, M.V., Joe, S.: An average discrepancy for optimal vertex-modified number-theoretic rules. Adv. Comput. Math. 12(1), 59–69 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford Science Publications, Oxford (1994)zbMATHGoogle Scholar
  15. 15.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex. 14(1), 1–33 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sloan, I.H., Reztsov, A.V.: Component-by-component construction of good lattice rules. Math. Comput. 71(237), 263–273 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sloan, I.H., Kuo, F.Y., Joe, S.: On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput. 71(240), 1609–1640 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ullrich, T.: Optimal cubature in Besov spaces with dominating mixed smoothness on the unit square. J. Complex. 30, 72–94 (2014)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceKU LeuvenLeuvenBelgium

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