Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quadrature Error Bounds with Applications to Lattice Rules
SIAM Journal on Numerical Analysis
A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension
Mathematics of Computation
Component-by-component construction of good lattice rules
Mathematics of Computation
The existence of good extensible rank-1 lattices
Journal of Complexity
Variance Reduction via Lattice Rules
Management Science
Finite-order weights imply tractability of multivariate integration
Journal of Complexity
Good Lattice Rules in Weighted Korobov Spaces with General Weights
Numerische Mathematik
Tractability of quasilinear problems I: General results
Journal of Approximation Theory
Infinite-Dimensional Quadrature and Approximation of Distributions
Foundations of Computational Mathematics
Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space
Journal of Computational and Applied Mathematics
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
Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN
Journal of Complexity
Journal of Complexity
Variance bounds and existence results for randomly shifted lattice rules
Journal of Computational and Applied Mathematics
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We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy, for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy-Krause variations over all lower-dimensional projections of the integrand. Under general conditions on the weights, we prove the existence of rank-1 lattice rules such that for any @d0, the general weighted star discrepancy is O(n^-^1^+^@d) for any number of points n1 (not necessarily prime), any shift of the lattice, general (decreasing) weights, and uniformly in the dimension. We also show that these rules can be constructed by a component-by-component strategy. This implies in particular that a single infinite-dimensional generating vector can be used for integrals in any number of dimensions, and even for infinite-dimensional integrands when they have bounded weighted variation. These same lattices are also good with respect to the worst-case error in weighted Korobov spaces with the same types of general weights. Similar results were already available for various special cases, such as general weights and prime n, or arbitrary n and product weights, but not for the most general combination of n composite, general weights, arbitrary shift, and star discrepancy, considered here. Our results imply tractability or strong tractability of integration for classes of integrands with finite weighted variation when the weights satisfy the conditions we give. These classes are a strict superset of those covered by earlier sufficient tractability conditions.