When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
Component-by-component construction of good lattice rules
Mathematics of Computation
Open problems for tractability of multivariate integration
Journal of Complexity
On the convergence rate of the component-by-component construction of good lattice rules
Journal of Complexity
Journal of Complexity
Quasi-Monte Carlo methods can be efficient for integration over products of spheres
Journal of Complexity
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
Randomly shifted lattice rules for unbounded integrands
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Lattice rule algorithms for multivariate approximation in the average case setting
Journal of Complexity
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
Time complexity estimation and optimisation of the genetic algorithm clustering method
WSEAS Transactions on Mathematics
Journal of Complexity
Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
Journal of Computational Physics
Weighted compound integration rules with higher order convergence for all N
Numerical Algorithms
Variance bounds and existence results for randomly shifted lattice rules
Journal of Computational and Applied Mathematics
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It is known from the analysis by Sloan and Woźniakowski that under appropriate conditions on the weights, the optimal rate of convergence for multivariate integration in weighted Korobov spaces is O(n-α/2+δ) where α 1 is some parameter of the spaces, δ is an arbitrary positive number, and the implied constant in the big O notation depends only on δ, and is independent on the number of variables. Similarly, the optimal rate for weighted Sobolev spaces is O(n-1+δ). However, their work did not show how to construct rules achieving these rates of convergence. The existing theory of the component-by-component constructions developed by Sloan, Kuo and Joe for the Sobolev case yields the rules achieving O(n-1/2) error bounds. Here we present theorems which show that those lattice rules constructed by the component-by-component algorithms in fact achieve the optimal rate of convergence under appropriate conditions on the weights.