A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Weighted tensor product algorithms for linear multivariate problems
Journal of Complexity
Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates
Mathematics and Computers in Simulation - IMACS sponsored Special issue on the second IMACS seminar on Monte Carlo methods
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
On the tractability of the Brownian bridge algorithm
Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
Finite-order weights imply tractability of multivariate integration
Journal of Complexity
Quasi-Monte Carlo methods can be efficient for integration over products of spheres
Journal of Complexity
Diaphony, discrepancy, spectral test and worst-case error
Mathematics and Computers in Simulation
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
Exact cubature for a class of functions of maximum effective dimension
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Complexity and effective dimension of discrete Lévy areas
Journal of Complexity
Low discrepancy sequences in high dimensions: How well are their projections distributed?
Journal of Computational and Applied Mathematics
Tractability properties of the weighted star discrepancy
Journal of Complexity
Comparison of Point Sets and Sequences for Quasi-Monte Carlo and for Random Number Generation
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications
Approximate zero-variance simulation
Proceedings of the 40th Conference on Winter Simulation
On the approximation error in high dimensional model representation
Proceedings of the 40th Conference on Winter Simulation
A practical view of randomized quasi-Monte Carlo: invited presentation, extended abstract
Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools
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
Error Estimates for the ANOVA Method with Polynomial Chaos Interpolation: Tensor Product Functions
SIAM Journal on Scientific Computing
Constructing adapted lattice rules using problem-dependent criteria
Proceedings of the Winter Simulation Conference
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A partial answer to why quasi-Monte Carlo (QMC) algorithms work well for multivariate integration was given in Sloan and Woźniakowski (J. Complexity 14 (1998) 1-33) by introducing weighted spaces. In these spaces the importance of successive coordinate directions is quantified by a sequence of weights. However, to be able to make use of weighted spaces for a particular application one has to make a choice of the weights. In this work, we take a more general view of the weights by allowing them to depend arbitrarily not only on the coordinates but also on the number of variables. Liberating the weights in this way allows us to give a recommendation for how to choose the weights in practice. This recommendation results from choosing the weights so as to minimize the error bound. We also consider how best to choose the underlying weighted Sobolev space within which to carry out the analysis. We revisit also lower bounds on the worst-case error, which change in many minor ways now, since the weights are allowed to depend on the number of variables, and we do not assume that the weights are uniformly bounded as has been assumed in previous papers. Necessary and sufficient conditions for QMC tractability and strong QMC tractability are obtained for the weighted Sobolev spaces with general weights. In the final section, we show that the analysis of variance decomposition of functions from one of the Sobolev spaces is equivalent to the decomposition of functions with respect to an orthogonal decomposition of this space.