Constructions of general polynomial lattice rules based on the weighted star discrepancy

In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration. Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy. In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy.

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

[2]  Robert F. Tichy,et al.  Sequences, Discrepancies and Applications , 1997 .

[3]  Dirk Nuyens,et al.  Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces , 2006, Math. Comput..

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

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

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

[7]  E. Haacke.  Sequences , 2005 .

[8]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[9]  Dirk Nuyens,et al.  Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points , 2006, J. Complex..

[10]  Harald Niederreiter,et al.  Monte Carlo and quasi-Monte Carlo methods 2004 , 2006 .

[11]  Harald Niederreiter,et al.  Optimal Polynomials for ( t,m,s )-Nets and Numerical Integration of Multivariate Walsh Series , 1996 .

[12]  Josef Dick,et al.  On Korobov Lattice Rules in Weighted Spaces , 2004, SIAM J. Numer. Anal..

[13]  Frances Y. Kuo,et al.  Reducing the construction cost of the component-by-component construction of good lattice rules , 2004, Math. Comput..

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

[15]  Frances Y. Kuo,et al.  Constructing Good Lattice Rules with Millions of Points , 2004 .

[16]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[17]  Josef Dick,et al.  Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces , 2005, J. Complex..

[18]  Dirk Nuyens,et al.  Fast Component-by-Component Construction, a Reprise for Different Kernels , 2006 .

[19]  Harald Niederreiter,et al.  Constructions of (t, m, s)-nets and (t, s)-sequences , 2005, Finite Fields Their Appl..

[20]  Ferenc Szidarovszky,et al.  Modeling uncertainty : an examination of stochastic theory, methods, and applications , 2002 .

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

[22]  G. Larcher.  Digital Point Sets: Analysis and Application , 1998 .

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

[24]  Josef Dick,et al.  The construction of extensible polynomial lattice rules with small weighted star discrepancy , 2007, Math. Comput..

[25]  Harald Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2002 , 2004 .

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

[27]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[28]  Deborah Monique,et al.  Authors' addresses , 2004 .

[29]  Pierre L'Ecuyer,et al.  Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation , 2001, SIAM J. Sci. Comput..

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

[31]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .