On the convergence rate of the component-by-component construction of good lattice rules

Authors:
Josef Dick
Affiliations:
School of Mathematics, University of New South Wales, Sydney 2052, Australia
Venue:
Journal of Complexity
Year:
2004

Citing 8
Cited 10

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
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces

SIAM Journal on Numerical Analysis
Component-by-component construction of good lattice rules with a composite number of points

Journal of Complexity
On the step-by-step construction of quasi: Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

Mathematics of Computation
Component-by-component construction of good lattice rules

Mathematics of Computation
Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces

Journal of Complexity
On Korobov Lattice Rules in Weighted Spaces

SIAM Journal on Numerical Analysis

Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces

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-Nyström method for Fredholm integral equations of the second kind with convolution type kernels

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 with the optimal rate of convergence for unbounded integrands

Journal of Complexity
Liberating the dimension

Journal of Complexity
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

Quantified Score

Hi-index	0.00

Visualization

Abstract

We prove error bounds on the worst-case error for integration in certain Korobov and Sobolev spaces using rank-1 lattice rules with generating vectors constructed by the component-by-component algorithm. For a prime number of points n a rate of convergence of the worst-case error for multivariate integration in Korobov spaces of O(n-x/2+δ), where α 1 is a parameter of the Korobov space and δ is an arbitrary positive real number, has been shown by Kuo. First we improve the constant of this error bound. Further, we prove an error bound which shows that the rate of convergence is optimal up to a power of log n for prime n. These error bounds are then generalised to the case where the number of points is not a prime number.Numerical results comparing the worst-case errors and the error bounds are presented.