The construction of extensible polynomial lattice rules with small weighted star discrepancy

. In this paper we introduce a construction algorithm for extensible polynomial lattice rules and we prove that the construction algorithm yields generating vectors of polynomials which are optimal for a range of moduli chosen in advance. The construction algorithm uses a sieve where the generating vectors are extended by one coefficient in each component at each step and where one keeps a certain number of good ones and discards the rest. We also show that the construction can be done component by component.

[1]  J HickernellF,et al.  Computing Multivariate Normal Probabilities Using Rank-1 Lattice Sequences , 1997 .

[2]  I. Sloan Lattice Methods for Multiple Integration , 1994 .

[3]  Fred J. Hickernell,et al.  Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature , 2000, SIAM J. Sci. Comput..

[4]  Harald Niederreiter,et al.  The Existence of Good Extensible Polynomial Lattice Rules , 2003 .

[5]  Fred J. Hickernell,et al.  The existence of good extensible rank-1 lattices , 2003, J. Complex..

[6]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[7]  Pierre L'Ecuyer,et al.  Recent Advances in Randomized Quasi-Monte Carlo Methods , 2002 .

[8]  Pierre L’Ecuyer,et al.  Polynomial Integration Lattices , 2004 .

[9]  Ian H. Sloan,et al.  Component-by-component construction of good lattice rules , 2002, Math. Comput..

[10]  Peter Kritzer,et al.  Constructions of general polynomial lattice rules based on the weighted star discrepancy , 2007, Finite Fields Their Appl..

[11]  I. H. SLOAN,et al.  Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces , 2002, SIAM J. Numer. Anal..

[12]  Harald Niederreiter,et al.  Low-discrepancy point sets obtained by digital constructions over finite fields , 1992 .

[13]  Frances Y. Kuo,et al.  Construction algorithms for polynomial lattice rules for multivariate integration , 2005, Math. Comput..

[14]  Josef Dick,et al.  Construction Algorithms for Digital Nets with Low Weighted Star Discrepancy , 2005, SIAM J. Numer. Anal..

[15]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[16]  S. Joe Component by Component Construction of Rank-1 Lattice Rules HavingO(n-1(In(n))d) Star Discrepancy , 2004 .

[17]  Frances Y. Kuo,et al.  Constructing Embedded Lattice Rules for Multivariate Integration , 2006, SIAM J. Sci. Comput..