Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Optimal Polynomials for (t,m,s)-Nets and Numerical Integration of Multivariate Walsh Series
SIAM Journal on Numerical Analysis
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation
SIAM Journal on Scientific Computing
The existence of good extensible rank-1 lattices
Journal of Complexity
On Korobov Lattice Rules in Weighted Spaces
SIAM Journal on Numerical Analysis
Construction Algorithms for Digital Nets with Low Weighted Star Discrepancy
SIAM Journal on Numerical Analysis
Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
Constructions of (t ,m,s)-nets and (t,s)-sequences
Finite Fields and Their Applications
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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.