Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

Abstract

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.