WAFOM on abelian groups for quasi-Monte Carlo point sets

TL;DR

The paper addresses provable-error QMC integration by extending the Dick weight and Walsh figure of merit ($\mathrm{WAFOM}$) to digital nets over arbitrary finite abelian groups $G$. It develops a $\mathrm{Koksma}$–$Hlawka$ type inequality for cyclic $G$, and proves a MacWilliams-type identity for the weight enumerator to enable efficient computation of $\mathrm{WAFOM}$ and the minimum Dick weight. It also provides lower and upper bounds on $\mathrm{WAFOM}$ as a function of the point-set size $N$, dimension $s$, and group size, generalizing known results from $G=\mathbb{F}_2$. These contributions yield existence results for low-$\mathrm{WAFOM}$ point sets and broaden higher-order QMC applicability beyond binary nets, with computable criteria for quality control.

Abstract

In this paper, we study quasi-Monte Carlo (QMC) rules for numerical integration. J. Dick proved a Koksma-Hlawka type inequality for $α$-smooth integrands and gave an explicit construction of QMC rules achieving the optimal rate of convergence in that function class. From this inequality, Matsumoto et al. introduced Walsh figure of merit (WAFOM) $\mathrm{WAFOM}(P)$ for an $\mathbb{F}_2$-digital net $P$ as a quickly computable quality criterion for $P$ as a QMC point set. The key ingredient for obtaining WAFOM is the Dick weight, a generalization of the Hamming weight and the Niederreiter-Rosenbloom-Tsfasman (NRT) weight. We extend the notions of the Dick weight and WAFOM for digital nets over a general finite abelian group $G$, and show that this version of WAFOM satisfies Koksma-Hlawka type inequality when $G$ is cyclic. We give a MacWilliams-type identity on the weight enumerator polynomials for the Dick weight, by which we can compute the minimum Dick weight as well as WAFOM. We give a lower bound of WAFOM of order $N^{-C'_G(\log N)/s}$ and an upper bound of lowest WAFOM of order $N^{-C_G(\log N)/s}$ for given $(G,N,s)$ if $(\log N)/s$ is sufficiently large, where $N$ is the cardinality of the point set $P$, $P$ is a quadrature rule in $[0,1)^s$, and $C'_G$ and $C_G$ are constants depending only on the cardinality of $G$. These bounds generalize the bounds given by Yoshiki and others given for $G=\mathbb{F}_2$.