# High-dimensional integration: The quasi-Monte Carlo way*†

```@article{Dick2013HighdimensionalIT,
title={High-dimensional integration: The quasi-Monte Carlo way*†},
author={Josef Dick and Frances Y. Kuo and Ian H. Sloan},
journal={Acta Numerica},
year={2013},
volume={22},
pages={133 - 288}
}```
• Published 2 April 2013
• Computer Science, Mathematics
• Acta Numerica
This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of…
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## References

SHOWING 1-10 OF 354 REFERENCES
QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND
• Mathematics
The ANZIAM Journal
• 2011
Abstract This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It
Quasi-Monte Carlo methods in finance
• P. L'Ecuyer
• Computer Science, Mathematics
Proceedings of the 2004 Winter Simulation Conference, 2004.
• 2004
We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, and the main classes of constructions underlying their
Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions
• J. Dick
• Mathematics, Computer Science
SIAM J. Numer. Anal.
• 2007
In this paper, we give explicit constructions of point sets in the \$s\$-dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case
When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?
• Computer Science, Mathematics
J. Complex.
• 1998
It is proved that the minimalworst case error of quasi-Monte Carlo algorithms does not depend on the dimensiondiff the sum of the weights is finite, and the minimal number of function values in the worst case setting needed to reduce the initial error by ? is bounded byC??p, where the exponentp? 1, 2], andCdepends exponentially on thesum of weights.
The effective dimension and quasi-Monte Carlo integration
• Computer Science, Mathematics
J. Complex.
• 2003
This paper analyses certain function classes commonly used in QMC methods for empirical and theoretical investigations and shows that the problem of determining their effective dimension is analytically tractable.
The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension
• Computer Science, Mathematics
Math. Comput.
• 2002
It is shown constructively using the Halton sequence that the e-exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.
Randomly shifted lattice rules for unbounded integrands
• Computer Science, Mathematics
J. Complex.
• 2006
It is proved that good randomly shifted lattice rules can be constructed component by component to achieve a worst case error of order O(n-1/2), where the implied constant can be independent of d.
Quasi-Monte Carlo Numerical Integration on Rs: Digital Nets and Worst-Case Error
• J. Dick
• Mathematics, Computer Science
SIAM J. Numer. Anal.
• 2011
It is proved that quasi-Monte Carlo-type rules for numerical integration of functions defined on \$\mathbb{R}^s\$ are optimal for numerical Integration in spaces of bounded fractional variation.
Finite-order weights imply tractability of multivariate integration
• Computer Science, Mathematics
J. Complex.
• 2004
This paper considers multivariate integration for the anchored and unanchored Sobolev spaces equipped with finite-order weights, and proves that classical low discrepancy sequences lead to error bounds with almost linear dependence on n-1 and polynomial dependence on d.
On Figures of Merit for Randomly-Shifted Lattice Rules
• Mathematics
• 2012
Randomized quasi-Monte Carlo (RQMC) can be seen as a variance reduction method that provides an unbiased estimator of the integral of a function f over the s-dimensional unit hypercube, with smaller