2014/11/29 by Makoto Matsumoto, Matsumoto, Makoto, Ryuichi Ohori +1
Materials Science · Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.1412.0168
openalex publication_date 2014/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix an integer s. Let f:[0,1)s → \mathbb R be an integrable function. Let P⊂ [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 α≥ 1, J. Dick gave a construction of point sets such that for α-smooth f, convergence rate O(N-α(log N)sα) 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-Clog N/s). WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of order N-α with α>1, for several test integrands for s=4 and 8.