Quasi-Monte Carlo for an Integrand with a Singularity Along a Diagonal in the Square



Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube [0, 1]d or at isolated possibly unknown points within [0, 1]d. Here we consider functions on the square [0, 1]2 that may become singular as the point approaches the diagonal line x1 = x2, and we study three quadrature methods. The first method splits the square into two triangles separated by a region around the line of singularity, and applies recently developed triangle QMC rules to the two triangular parts. For functions with a singularity ‘no worse than |x1 − x2|A is’ for 0 < A < 1 that method yields an error of \(O( (\log (n)/n)^{(1-A)/2})\). We also consider methods extending the integrand into a region containing the singularity and show that method will not improve upon using two triangles. Finally, we consider transforming the integrand to have a more QMC-friendly singularity along the boundary of the square. This then leads to error rates of O(n−1+𝜖+A) when combined with some corner-avoiding Halton points or with randomized QMC but it requires some stronger assumptions on the original singular integrand.



This work was supported by the US National Science Foundation under grants DMS-1407397 and DMS-1521145. We thank two anonymous reviewers for helpful comments and we thank Frances Kuo for her work handling our paper.


  1. 1.
    Basu, K., Owen, A.B.: Low discrepancy constructions in the triangle. SIAM J. Num. An. 53(2), 743–761 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Basu, K., Owen, A.B.: Scrambled geometric net integration over general product spaces. Found. Comput. Math. 17(2), 1–30 (2015)MathSciNetGoogle Scholar
  3. 3.
    Binder, C.: Über einen Satz von de Bruijn und Post. Öst. Akad. der Wiss. Math.-Natur. Klasse. Sitz. Abteilung II 179, 233–251 (1970)zbMATHGoogle Scholar
  4. 4.
    Brandolini, L., Colzani, L., Gigante, G., Travaglini, G.: A Koksma–Hlawka inequality for simplices. In: Trends in Harmonic Analysis, pp. 33–46. Springer, Berlin (2013)CrossRefGoogle Scholar
  5. 5.
    de Bruijn, N.G., Post, K.A.: A remark on uniformly distributed sequences and Riemann integrability. Indag. Math. 30, 149–150 (1968)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dick, J., Pillichshammer, F.: Digital Sequences, Discrepancy and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  7. 7.
    Klinger, B.: Discrepancy of point sequences and numerical integration. Ph.D. thesis, Technische Universität Graz (1997)Google Scholar
  8. 8.
    Klinger, B.: Numerical integration of singular integrands using low-discrepancy sequences. Computing 59, 223–236 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mishra, M., Gupta, N.: Application of quasi Monte Carlo integration technique in EM scattering from finite cylinders. Prog. Electromagn. Res. Lett. 9, 109–118 (2009)CrossRefGoogle Scholar
  10. 10.
    Owen, A.B.: Monte Carlo variance of scrambled equidistribution quadrature. SIAM J. Numer. Anal. 34(5), 1884–1910 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Owen, A.B.: Multidimensional variation for quasi-Monte Carlo. In: Fan, J., Li, G. (eds.) International Conference on Statistics in Honour of Professor Kai-Tai Fang’s 65th Birthday (2005)CrossRefGoogle Scholar
  12. 12.
    Owen, A.B.: Halton sequences avoid the origin. SIAM Rev. 48, 487–583 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Owen, A.B.: Quasi-Monte Carlo for integrands with point singularities at unknown locations. In: Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 403–417. Springer, Berlin (2006)Google Scholar
  14. 14.
    Sobol’, I.M.: Calculation of improper integrals using uniformly distributed sequences. Sov. Math. Dokl. 14(3), 734–738 (1973)Google Scholar
  15. 15.
    Wang, X., Sloan, I.H.: Quasi-Monte Carlo methods in financial engineering: an equivalence principle and dimension reduction. Oper. Res. 59(1), 80–95 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LinkedIn Inc.Mountain ViewUSA
  2. 2.Stanford UniversityStanfordUSA

Personalised recommendations