Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Variance Reduction via Lattice Rules
Management Science
On the convergence rate of the component-by-component construction of good lattice rules
Journal of Complexity
Good Lattice Rules in Weighted Korobov Spaces with General Weights
Numerische Mathematik
Efficient Weighted Lattice Rules with Applications to Finance
SIAM Journal on Scientific Computing
Constructing Embedded Lattice Rules for Multivariate Integration
SIAM Journal on Scientific Computing
Staffing Multiskill Call Centers via Linear Programming and Simulation
Management Science
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
Constructing adapted lattice rules using problem-dependent criteria
Proceedings of the Winter Simulation Conference
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We study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions. We consider integrands that are square integrable but whose Fourier series are not necessarily absolutely convergent. For such integrands, a bound on the variance is expressed through a certain type of weighted discrepancy. We prove existence and construction results for randomly shifted lattice rules such that the variance bounds are almost O(n^-^@a), where n is the number of function evaluations and @a1 depends on our assumptions on the convergence speed of the Fourier coefficients. These results hold for general weights, arbitrary n, and any dimension. With additional conditions on the weights, we obtain a convergence that holds uniformly in the dimension, and this provides sufficient conditions for strong tractability of the integration problem. We also show that lattice rules that satisfy these bounds are not difficult to construct explicitly and we provide numerical illustrations of the behaviour of construction algorithms.