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

Abstract

Fix an integer s. Let $f:[0,1)^s \rightarrow {\mathbb {R}}$ be an integrable function. Let ${\mathscr {P}}\subset [0, 1]^s$ be a finite point set. Quasi-Monte Carlo integration of f by ${\mathscr {P}}$ is the average value of f over ${\mathscr {P}}$ that approximates the integration of f over the s-dimensional cube. Koksma–Hlawka inequality tells that, by a smart choice of ${\mathscr {P}}$, one may expect that the error decreases roughly $O(N^{-1}(\log N)^s)$. For any $\alpha \ge 1$, 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.