The existence of good extensible rank-1 lattices

Fred J. Hickernell; Harald Niederreiter

doi:10.1016/S0885-064X(02)00026-2

The existence of good extensible rank-1 lattices

Fred J. Hickernell^*, Harald Niederreiter

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › peer-review

60 Citations (Scopus)

Abstract

Extensible integration lattices have the attractive property that the number of points in the node set may be increased while retaining the existing points. It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b², ... points and dimensions s = 1, 2, ... the figures of merit R_α, P_α, and discrepancy are all small. The upper bounds obtained on these figures of merit for extensible lattices are some power of log n worse than the best upper bounds for lattices where h is allowed to vary with n and s.

Original language	English
Pages (from-to)	286-300
Number of pages	15
Journal	Journal of Complexity
Volume	19
Issue number	3
DOIs	https://doi.org/10.1016/S0885-064X(02)00026-2
Publication status	Published - Jun 2003
Externally published	Yes

Scopus Subject Areas

Algebra and Number Theory
Statistics and Probability
Numerical Analysis
General Mathematics
Control and Optimization
Applied Mathematics

User-Defined Keywords

Discrepancy
Extensible lattices
Figures of merit
Lattice rules
Quasi-Monte Carlo integration

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10.1016/S0885-064X(02)00026-2

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title = "The existence of good extensible rank-1 lattices",

abstract = "Extensible integration lattices have the attractive property that the number of points in the node set may be increased while retaining the existing points. It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b2, ... points and dimensions s = 1, 2, ... the figures of merit Rα, Pα, and discrepancy are all small. The upper bounds obtained on these figures of merit for extensible lattices are some power of log n worse than the best upper bounds for lattices where h is allowed to vary with n and s.",

keywords = "Discrepancy, Extensible lattices, Figures of merit, Lattice rules, Quasi-Monte Carlo integration",

author = "Hickernell, {Fred J.} and Harald Niederreiter",

note = "Funding Information: This work was partially supported by a Hong Kong Research Grants Council Grant HKBU/2030/99P. ",

year = "2003",

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language = "English",

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T1 - The existence of good extensible rank-1 lattices

AU - Hickernell, Fred J.

AU - Niederreiter, Harald

N1 - Funding Information: This work was partially supported by a Hong Kong Research Grants Council Grant HKBU/2030/99P.

PY - 2003/6

Y1 - 2003/6

N2 - Extensible integration lattices have the attractive property that the number of points in the node set may be increased while retaining the existing points. It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b2, ... points and dimensions s = 1, 2, ... the figures of merit Rα, Pα, and discrepancy are all small. The upper bounds obtained on these figures of merit for extensible lattices are some power of log n worse than the best upper bounds for lattices where h is allowed to vary with n and s.

AB - Extensible integration lattices have the attractive property that the number of points in the node set may be increased while retaining the existing points. It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b2, ... points and dimensions s = 1, 2, ... the figures of merit Rα, Pα, and discrepancy are all small. The upper bounds obtained on these figures of merit for extensible lattices are some power of log n worse than the best upper bounds for lattices where h is allowed to vary with n and s.

KW - Discrepancy

KW - Extensible lattices

KW - Figures of merit

KW - Lattice rules

KW - Quasi-Monte Carlo integration

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U2 - 10.1016/S0885-064X(02)00026-2

DO - 10.1016/S0885-064X(02)00026-2

M3 - Journal article

AN - SCOPUS:0037648331

SN - 0885-064X

VL - 19

SP - 286

EP - 300

JO - Journal of Complexity

JF - Journal of Complexity

IS - 3

ER -

The existence of good extensible rank-1 lattices

Abstract

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