Foundations of Computational Mathematics

, Volume 15, Issue 5, pp 1245–1278 | Cite as

Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order



Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced by Dick (Ann Stat 39:1372–1398, 2011) and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the \(s\)-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is \(ds\), where the integer \(d\) is the so-called interlacing factor. In this paper we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen’s full scrambling scheme as well as the simplifications studied by Hickernell, Matoušek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to \(\alpha \ge 1\), and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base \(b\) and dimension \(ds\), to which we then apply the interlacing scheme of order \(d\), we obtain a construction cost of the algorithm of order \(\mathcal {O}(dsmb^m)\) operations using \(\mathcal {O}(b^m)\) memory in case of product weights, where \(b^m\) is the number of points in the polynomial lattice point set.


Randomized quasi-Monte Carlo Interlaced scrambled polynomial lattice rules Component-by-component construction  Weighted function spaces Tractability of multivariate integration 

Mathematics Subject Classification

Primary 65C05 Secondary 65D30, 65D32 



The first author is supported by JSPS Grant-in-Aid for JSPS Fellows No. 24-4020, and the second author is supported by a Queen Elizabeth 2 Fellowship (DP1097023) from the Australian Research Council. T.G. would like to thank Josef Dick for his hospitality while visiting the University of New South Wales where this research was carried out.


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Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Graduate School of EngineeringThe University of TokyoTokyoJapan
  2. 2.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia

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