Lattice methods for multiple integration: theory, error analysis and examples
SIAM Journal on Numerical Analysis
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
The existence of good extensible rank-1 lattices
Journal of Complexity
Quasi-Monte Carlo algorithms for unbounded, weighted integration problems
Journal of Complexity
Good Lattice Rules in Weighted Korobov Spaces with General Weights
Numerische Mathematik
Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
Randomly shifted lattice rules for unbounded integrands
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Constructing Embedded Lattice Rules for Multivariate Integration
SIAM Journal on Scientific Computing
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We approximate weighted integrals over Euclidean space by using shifted rank-1 lattice rules with good bounds on the ''generalised weighted star discrepancy''. This version of the discrepancy corresponds to the classic L"~ weighted star discrepancy via a mapping to the unit cube. The weights here are general weights rather than the product weights considered in earlier works on integrals over R^d. Known methods based on an averaging argument are used to show the existence of these lattice rules, while the component-by-component technique is used to construct the generating vector of these shifted lattice rules. We prove that the bound on the weighted star discrepancy considered here is of order O(n^-^1^+^@d) for any @d0 and with the constant involved independent of the dimension. This convergence rate is better than the O(n^-^1^/^2) achieved so far for both Monte Carlo and quasi-Monte Carlo methods.