A construction of polynomial lattice rules with small gain coefficients
- First Online:
- 104 Downloads
In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Owen, indicate to what degree the method improves upon Monte Carlo. We show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component-by-component decays at a rate of N−(2α+1)+δ, for all δ > 0, assuming that the function under consideration has bounded variation of order α for some 0 < α ≤ 1, and where N denotes the number of quadrature points. An analogous result is obtained for Korobov polynomial lattice rules. It is also established that these rules are almost optimal for the function space considered in this paper. Furthermore, we discuss the implementation of the component-by-component approach and show how to reduce the computational cost associated with it. Finally, we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets.
Mathematics Subject Classification (2000)65C05 65D30 65D32
Unable to display preview. Download preview PDF.
- 1.Baldeaux, J.: Higher order nets and sequences, PhD thesis, The University of New South Wales (2010)Google Scholar
- 2.Caflish R.E., Morokoff W.J., Owen A.B.: Valuation of mortgage backed securities using Brownian Bridges to reduce effective dimension. J. Comput. Finance 1, 27–46 (1997)Google Scholar
- 3.Dick, J.: On the fast component-by-component algorithm for polynomial lattice rules. http://quasirandomideas.wordpress.com/2009/12/31/fast-cbc-for-polynomial-lattice-rules, Posted on December 31st 2009, Last accessed February 2nd 2010
- 4.Dick, J., Gnewuch, M.: Embedding theorems for fractional spaces and numerical integration (in preparation)Google Scholar
- 7.Dick, J., Pillichshammer, F.: Digital nets and sequences, discrepancy and quasi-monte carlo integration. Cambridge University Press, Cambridge (to appear, 2010)Google Scholar
- 18.Matoušek J.: Geometric Discrepancy, Algorithms and Combinatorics, vol. 18. Springer, Berlin (1999)Google Scholar
- 20.Niederreiter, H.: Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)Google Scholar
- 22.Novak E.: Deterministic and stochastic error bounds in numerical analysis. Lecture Notes in Mathematics, vol. 1349. Springer, Berlin (1988)Google Scholar
- 25.Owen A.B.: Randomly permuted (t,m,s)-nets and (t,s)-sequences. In: Niederreiter, H., Jau-Shyong Shiue, P. (eds) Monte Carlo and quasi-Monte Carlo Methods in Scientific Computing, pp. 299–317. Springer, New York (1995)Google Scholar