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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.