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

摘要

The $\mathcal{L}_2$ discrepancy is one of several well-known quantitativemeasures for the equidistribution properties of point sets in thehigh-dimensional unit cube. The concept of weights was introduced by Sloan andWo\'{z}niakowski to take into account the relative importance of thediscrepancy of lower dimensional projections. As known under the name ofquasi-Monte Carlo methods, point sets with small weighted $\mathcal{L}_2$discrepancy are useful in numerical integration. This study investigates thecomponent-by-component construction of polynomial lattice rules over the finitefield $\mathbb{F}_2$ 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 experimentsconfirm that the performance of our constructed polynomial lattice point setsis comparable or even superior to that of Sobol' sequences.