Parallel Quasi-Monte Carlo Integration by Partitioning Low Discrepancy Sequences

  • Alexander Keller
  • Leonhard Grünschloß
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)


A general concept for parallelizing quasi-Monte Carlo methods is introduced. By considering the distribution of computing jobs across a multiprocessor as an additional problem dimension, the straightforward application of quasi-Monte Carlo methods implies parallelization. The approach in fact partitions a single low-discrepancy sequence into multiple low-discrepancy sequences. This allows for adaptive parallel processing without synchronization, i.e. communication is required only once for the final reduction of the partial results. Independent of the number of processors, the resulting algorithms are deterministic, and generalize and improve upon previous approaches.


Adaptive Sampling Radical Inverse Integer Base Halton Sequence Inverse Permutation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work has been dedicated to Stefan Heinrich’s 60th birthday. The authors thank Matthias Raab for discussion.


  1. 1.
    Abramov, G.: US patent #6,911,976: System and method for rendering images using a strictly-deterministic methodology for generating a coarse sequence of sample points (2002)Google Scholar
  2. 2.
    Bromley, B.: Quasirandom number generators for parallel Monte Carlo algorithms. J. Parallel Distrib. Comput. 38(1), 101–104 (1996)Google Scholar
  3. 3.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press (2009)Google Scholar
  4. 4.
    Entacher, K., Schell, T., Schmid, W., Uhl, A.: Defects in parallel Monte Carlo and quasi-Monte Carlo integration using the leap-frog technique. Parallel Algorithms Appl. pp. 13–26 (2003)Google Scholar
  5. 5.
    Faure, H.: Good permutations for extreme discrepancy. J. Number Theory 42, 47–56 (1992)Google Scholar
  6. 6.
    Grünschloß, L.: Motion Blur. Master’s thesis, Universität Ulm (2008)Google Scholar
  7. 7.
    Hickernell, F., Hong, H., L’Ecuyer, P., Lemieux, C.: Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J. Sci. Comput. 22, 1117–1138 (2000)Google Scholar
  8. 8.
    Jensen, H.: Realistic Image Synthesis Using Photon Mapping. AK Peters (2001)Google Scholar
  9. 9.
    Jez, P., Uhl, A., Zinterhof, P.: Applications and parallel implementation of QMC integration. In: R. Trobec, M. Vajteršic, P. Zinterhof (eds.) Parallel Computing, pp. 175–215. Springer (2008)Google Scholar
  10. 10.
    Keller, A.: Myths of computer graphics. In: H. Niederreiter (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 217–243. Springer (2006)Google Scholar
  11. 11.
    Keller, A., Grünschloß, L., Droske, M.: Quasi-Monte Carlo progressive photon mapping. In: L. Plaskota, H. Woźniakowski (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 501–511. Springer (2012)Google Scholar
  12. 12.
    Kocis, L., Whiten, W.: Computational investigations of low-discrepancy sequences. ACM Trans. Math. Softw. 23(2), 266–294 (1997)Google Scholar
  13. 13.
    Kollig, T., Keller, A.: Efficient multidimensional sampling. Computer Graphics Forum (Proc. Eurographics 2002) 21(3), 557–563 (2002)Google Scholar
  14. 14.
    Larcher, G., Pillichshammer, F.: Walsh series analysis of the L 2-discrepancy of symmetrisized point sets. Monatsh. Math. 132, 1–18 (2001)Google Scholar
  15. 15.
    Matoušek, J.: On the L 2-discrepancy for anchored boxes. J. Complexity 14(4), 527–556 (1998)Google Scholar
  16. 16.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)Google Scholar
  17. 17.
    Ökten, G., Srinivasan, A.: Parallel quasi-Monte Carlo methods on a heterogeneous cluster. In: K.T. Fang, F. Hickernell, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 406–421. Springer (2002)Google Scholar
  18. 18.
    Schmid, W., Uhl, A.: Parallel quasi-Monte Carlo integration using (t, s)-sequences. In: ParNum ’99: Proceedings of the 4th International ACPC Conference Including Special Tracks on Parallel Numerics and Parallel Computing in Image Processing, Video Processing, and Multimedia, pp. 96–106. Springer-Verlag, London, UK (1999)Google Scholar
  19. 19.
    Schmid, W., Uhl, A.: Techniques for parallel quasi-Monte Carlo integration with digital sequences and associated problems. Mathematics and Computers in Simulation 55(1–3), 249–257 (2001)Google Scholar
  20. 20.
    Schwarz, H., Köckler, N.: Numerische Mathematik. 6. überarb. Auflage, Vieweg + Teubner (2008)Google Scholar
  21. 21.
    Sobol’, I.: On the Distribution of points in a cube and the approximate evaluation of integrals. Zh. vychisl. Mat. mat. Fiz. 7(4), 784–802 (1967)Google Scholar
  22. 22.
    Wächter, C.: Quasi-Monte Carlo Light Transport Simulation by Efficient Ray Tracing. Ph.D. thesis, Universität Ulm (2008)Google Scholar
  23. 23.
    Zaremba, S.: La discrépance isotrope et l’intégration numérique. Ann. Mat. Pura Appl. 87, 125–136 (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.NVIDIA ARC GmbHBerlinGermany
  2. 2.Rendering Research, Weta DigitalWellingtonNew Zealand

Personalised recommendations