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

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 $8$.