Numerische Mathematik

, Volume 141, Issue 2, pp 399–427 | Cite as

Lattice rules in non-periodic subspaces of Sobolev spaces

  • Takashi GodaEmail author
  • Kosuke Suzuki
  • Takehito Yoshiki


We investigate quasi-Monte Carlo (QMC) integration over the s-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {sob}})\) and their subspaces of high order smoothness \(\alpha >1\), where \(\varvec{\gamma }\) denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter \(\alpha >1/2\) consisting of non-periodic smooth functions, denoted by \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {cos}})\), and also in the sum of half-period cosine spaces and Korobov spaces with common parameter \(\alpha \), denoted by \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {kor}+\mathrm {cos}})\). Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer \(\alpha \), we provide their corresponding norm-equivalent subspaces of \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {sob}})\). This implies that \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {kor}+\mathrm {cos}})\) is strictly smaller than \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {sob}})\) as sets for \(\alpha \ge 2\), which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in \(\mathcal {H}\left( K_{2,\varvec{\gamma },s}^{\mathrm {sob}}\right) \) and also the worst-case error of symmetrized lattice rules in an intermediate space between \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {kor}+\mathrm {cos}})\) and \(\mathcal {H}(K_{\alpha ,\varvec{\gamma },s}^{\mathrm {sob}})\). We show that the almost optimal rate of convergence can be achieved for both cases, while a weak dependence of the worst-case error bound on the dimension can be obtained for the former case.

Mathematics Subject Classification

65C05 65D30 65D32 



The authors would like to thank the anonymous reviewers for their suggestions and comments. This work was supported by JSPS Grant-in-Aid for Young Scientists No. 15K20964 (T. G.), JSPS Grant-in-Aid for JSPS Fellows Nos. 17J00466 (K. S.) and 17J02651 (T. Y.), and JST CREST.


  1. 1.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cools, R., Kuo, F.Y., Nuyens, D., Suryanarayana, G.: Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions. J. Complex. 36, 166–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dick, J.: On the convergence rate of the component-by-component construction of good lattice rules. J. Complex. 20, 493–522 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dick, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dick, J.: How to generate higher order Sobol points in Matlab and some numerical examples. (2010). Accessed 24 July 2018
  6. 6.
    Dick, J., Goda, T., Yoshiki, T.: Richardson extrapolation of polynomial lattice rules. ArXiv preprint, arXiv:1707.03989
  7. 7.
    Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dick, J., Nuyens, D., Pillichshammer, F.: Lattice rules for nonperiodic smooth integrands. Numer. Math. 126, 259–291 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Good lattice rules in weighted Korobov spaces with general weights. Numer. Math. 103, 63–97 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goda, T., Suzuki, K., Yoshiki, T.: Optimal order quadrature error bounds for infinite-dimensional higher-order digital sequences. Found. Comput. Math. 18, 433–458 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, Cambridge (2007)zbMATHGoogle Scholar
  13. 13.
    Hickernell, F.J.: Obtaining \(O(N^{-2+\epsilon })\) convergence for lattice quadrature rules. In: Fang, K.T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 274–289. Springer, Berlin (2002)CrossRefGoogle Scholar
  14. 14.
    Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. J. Complex. 19, 301–320 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond. ANZIAM J. 53, 1–37 (2011). (corrected in 54 (2013) 216–219)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992)CrossRefGoogle Scholar
  17. 17.
    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume I: Linear Information, EMS Tracts in Mathematics, vol. 6. European Mathematical Society (EMS), Zürich (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sloan, I.H., Joe, S.: Lattice Methods for Multivariate Integration. Oxford University Press, Oxford (1994)zbMATHGoogle Scholar
  20. 20.
    Sloan, I.H., Reztsov, A.V.: Component-by-component construction of good lattice rules. Math. Comput. 71, 263–273 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complex. 14, 1–33 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sloan, I.H., Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Complex. 17, 697–721 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of EngineeringUniversity of TokyoTokyoJapan
  2. 2.Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
  3. 3.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

Personalised recommendations