Walsh Figure of Merit for Digital Nets: An Easy Measure for Higher Order Convergent QMC

TL;DR

This paper introduces Walsh figure of Merit (WAFOM) as an easy-to-compute measure for evaluating the quality of digital nets in quasi-Monte Carlo methods, demonstrating its effectiveness in achieving higher convergence rates.

Contribution

The paper presents WAFOM as a practical and efficient measure for assessing digital nets, enabling better construction of point sets with higher convergence rates in QMC integration.

Findings

WAFOM effectively predicts convergence rates for digital nets.

Low WAFOM point sets achieve convergence rates faster than traditional bounds.

Empirical results show convergence of order N^{-eta} with eta > 1.

Abstract

Fix an integer $s$. Let $f:[0,1)^s \to \mathbb R$ be an integrable function. Let $P\subset [0,1]^s$ be a finite point set. Quasi-Monte Carlo integration of $f$ by $P$ is the average value of $f$ over $P$ that approximates the integration of $f$ over the $s$-dimensional cube. Koksma-Hlawka inequality tells that, by a smart choice of $P$, one may expect that the error decreases roughly $O(N^{-1}(\log N)^s)$. For any $\alpha \geq 1$, J.\ Dick gave a construction of point sets such that for $\alpha$-smooth $f$, convergence rate $O(N^{-\alpha}(\log N)^{s\alpha})$ is assured. As a coarse version of his theory, M-Saito-Matoba introduced Walsh figure of Merit (WAFOM), which gives the convergence rate $O(N^{-C\log N/s})$. WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of order $N^{-\alpha}$ with $\alpha>1$, for several test integrands for $s=4$ and $8$.