Title: Polynomial copy rules in Walsh spaces
Authors: Nuyens, Dirk ×
Cools, Ronald #
Issue Date: Jun-2008
Conference: Foundations of Computational Mathematics edition:2008 location:City University of Hong Kong date:16-26 June 2008
Abstract: In the 1990's copy rules have been studied as an easy and convenient way to construct lattice rules with a huge amount
of points.
Starting with the analysis of higher-rank lattice rules it was demonstrated numerically by Sloan and Walsh (1990) that
copying scaled down rules by $\tfrac{1}{2}$ or $\tfrac{1}{3}$th to fill up the unit cube---conveniently only in the f
irst few dimensions to avoid a curse of dimensionality by construction---could deliver rules which have a better worst
-case error for the standard Korobov space than basic rank-$1$ rules.
Further results were obtained by Disney and Sloan (1992), Joe and Sloan (1992) and Joe and Disney (1993).
More recently, these results were extended to the product weighted Korobov space by Kuo and Joe (2003) in which condit
ions on the weights were found to obtain a smaller worst-case error for an (intermediate) copy rule than for a rank-$1
$ rule.
(Another recent track, called augmentation sequences, is investigated
by Robinson, Li and Hill, but these are not directly relevant to this talk.)

Here we present results on polynomial copy rules in a weighted Walsh space.
We show some similarities with normal copy rules, as can be expected by
the similarities between polynomial lattice rules and normal lattice
rules, and some differences which show up in the theory.
We derive an expression for the mean of the worst-case error and an
existence theorem for polynomial copy rules which will do better than
the mean follows from this. Comparing this mean to the mean of normal polynomial
lattice rules, we can analyse when copying is expected to be favorable.
We remark that, as is the case for normal copy rules, polynomial copy
rules can also be constructed by the fast component-by-component construction algorithm.
Publication status: published
KU Leuven publication type: IMa
Appears in Collections:Numerical Analysis and Applied Mathematics Section
× corresponding author
# (joint) last author

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