Implementation of a Component-By-Component Algorithm to Generate Small Low-Discrepancy Samples
In [B. Doerr, M. Gnewuch, P. Kritzer, F. Pillichshammer. Monte Carlo Methods Appl., 14:129–149, 2008], a component-by-component (CBC) approach to generate small low-discrepancy samples was proposed and analyzed. The method is based on randomized rounding satisfying hard constraints and its derandomization. In this paper we discuss how to implement the algorithm and present first numerical experiments. We observe that the generated points in many cases have a significantly better star discrepancy than what is guaranteed by the theoretical upper bound. Moreover, we exhibit that the actual discrepancy is mainly caused by the underlying grid structure, whereas the rounding errors have a negligible contribution. Hence to improve the algorithm, we propose and analyze a randomized point placement. We also study a hybrid approach which combines classical low-discrepancy sequences and the CBC algorithm.
KeywordsMonte Carlo Failure Probability Star Discrepancy Placement Error Cache Method
Unable to display preview. Download preview PDF.
- 1.Doerr, B., Gnewuch, M.: Construction of low-discrepancy point sets of small size by bracketing covers and dependent randomized rounding. In: A. Keller, S. Heinrich, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 299–312. Springer-Verlag, Berlin Heidelberg (2008) CrossRefGoogle Scholar
- 4.Doerr, B., Wahlström, M.: Randomized rounding in the presence of a cardinality constraint. In: Proceedings of ALENEX, pp. 162–174. SIAM (2009) Google Scholar
- 15.Spanier, J.: Quasi-Monte Carlo methods for particle transport problems. In: H. Niederreiter, P.J.S. Shiue (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, pp. 121–148. Springer-Verlag, Berlin (1995) Google Scholar