Paper Info
Title
High-dimensional integration: The quasi-Monte Carlo way
Abstract
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 convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called 'weights', since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.
Year
DOI
Venue
2013
10.1017/S0962492913000044
Acta Numerica
Field
DocType
Volume
Kernel (linear algebra),Hilbert space,Function space,Mathematical optimization,Mathematical analysis,Level of detail,Computer science,Quasi-Monte Carlo method,Rate of convergence,Unit cube,Bounded function
Journal
22
ISSN
Citations 
PageRank 
0962-4929
70
3.18
References 
Authors
86
3
Name
Order
Citations
PageRank
Josef Dick155756.49
Frances Y. Kuo247945.19
Ian H. Sloan31180183.02