Abstract
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 x 1 = x 2, 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 |x 1 − x 2|−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.
Dedicated to Ian H. Sloan on the occasion of his 80th birthday.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Basu, K., Owen, A.B.: Low discrepancy constructions in the triangle. SIAM J. Num. An. 53(2), 743–761 (2015)
Basu, K., Owen, A.B.: Scrambled geometric net integration over general product spaces. Found. Comput. Math. 17(2), 1–30 (2015)
Binder, C.: Über einen Satz von de Bruijn und Post. Öst. Akad. der Wiss. Math.-Natur. Klasse. Sitz. Abteilung II 179, 233–251 (1970)
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)
de Bruijn, N.G., Post, K.A.: A remark on uniformly distributed sequences and Riemann integrability. Indag. Math. 30, 149–150 (1968)
Dick, J., Pillichshammer, F.: Digital Sequences, Discrepancy and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)
Klinger, B.: Discrepancy of point sequences and numerical integration. Ph.D. thesis, Technische Universität Graz (1997)
Klinger, B.: Numerical integration of singular integrands using low-discrepancy sequences. Computing 59, 223–236 (1997)
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)
Owen, A.B.: Monte Carlo variance of scrambled equidistribution quadrature. SIAM J. Numer. Anal. 34(5), 1884–1910 (1997)
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)
Owen, A.B.: Halton sequences avoid the origin. SIAM Rev. 48, 487–583 (2006)
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)
Sobol’, I.M.: Calculation of improper integrals using uniformly distributed sequences. Sov. Math. Dokl. 14(3), 734–738 (1973)
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)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Basu, K., Owen, A.B. (2018). Quasi-Monte Carlo for an Integrand with a Singularity Along a Diagonal in the Square. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-72456-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72455-3
Online ISBN: 978-3-319-72456-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)