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.