Construction of scrambled polynomial lattice rules over $\mathbb{F}_2$ with small mean square weighted $\mathcal{L}_2$ discrepancy

TL;DR

This paper develops a method to construct scrambled polynomial lattice rules over _2 with low mean square weighted _2 discrepancy, improving numerical integration in high dimensions.

Contribution

It introduces a component-by-component construction for polynomial lattice rules with provably small discrepancy, achieving near-optimal convergence rates.

Findings

Discrepancy bounds converge at nearly the best possible rate of N^{-2+δ}.

Constructed point sets perform comparably or better than Sobol' sequences.

Numerical experiments validate the theoretical discrepancy bounds.

Abstract

The $\mathcal{L}_2$ discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Wo\'{z}niakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted $\mathcal{L}_2$ discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field $\mathbb{F}_2$ whose scrambled point sets have small mean square weighted $\mathcal{L}_2$ discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of $N^{-2+\delta}$ for all $\delta>0$, where $N$ denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol' sequences.