Recently J. Baldeaux and myself submitted the manuscript titled A Construction of Polynomial Lattice Rules with Small Gain Coefficients.
In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Art Owen, indicate to what degree the method improves upon Monte Carlo. We show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component-by-component decays at a rate of , for all , assuming that the function under consideration bounded fractional variation of order and where denotes the number of quadrature points. We give some further comments on the paper.
where are a randomly scrambled polynomial lattice point set. Since random scrambling yields an unbiased estimator one has
Hence one considers the variance of the estimator
It is known that
The term depends only on the function , whereas the term depends only on the polynomial lattice (here is the dual lattice which has generating vector and modulus ). We define the function norm
(We also show that if has bounded fractional variation of order then .) Hence one can estimate
Using this bound in Equation (1) one obtains
We show that
can be used as quality criterion in a component-by-component construction of polynomial lattices. We need to assume that , as for $\alpha = 0$ one has . We show that we obtain optimal upper bounds on the variance with this approach. Further, if one constructs a polynomial lattice rule for some , then this polynomial lattice rule will also be optimal if the function has smoothness with . Unfortunately our method can only be used for and hence the boundary case is excluded.
It would be interesting to also obtain a result for . One may proceed in the following way. To obtain Equation (2) from Equation (1) we used Hölder's inequality. Hence using Hölder's inequality with different parameters one can obtain a function norm and bound in norm . Indeed, if one uses the -norm for and the -norm for one obtains a function which is also finite for . One could then use this bound as quality criterion in a component-by-component approach. Although interesting, we did not find a closed form expression for this bound and hence we were not able to use the component-by-component approach. In fact, the only case where we could find a closed form expression for which we used in the paper and which we outlined above.