Construction of interlaced scrambled polynomial lattice rules of arbitrary high order

TL;DR

This paper develops a method to construct high-order scrambled polynomial lattice rules for quasi-Monte Carlo integration, improving dimension dependence and achieving optimal convergence rates for smooth functions.

Contribution

It introduces a construction of interlaced scrambled polynomial lattice rules with explicit algorithms, enhancing dimension dependence while maintaining optimal convergence.

Findings

Achieves optimal convergence rate for smooth functions

Provides explicit component-by-component construction algorithms

Improves dimension dependence in quasi-Monte Carlo methods

Abstract

Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011), 1372--1398] 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, Matou\v{s}ek 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.