2015/04/10 by Takehito Yoshiki, Yoshiki, Takehito
Mathematics · #65D30 #65D32 #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1504.03175
openalex publication_date 2015/04/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function f\colon[0,1)s→ ℝ by a finite point set P⊂ [0,1)s is the approximation of the integral I(f):=∫[0,1)sf(x) dx by the average IP(f):=(1)/(|P|)∑x ∈ Pf(x). We treat a certain class of point sets P called digital nets. A Koksma-Hlawka type inequality is an inequality bounding the integration error Err(f;P):=I(f)-IP(f) by a bound of the form |Err(f;P)|≤ C⋅ ‖f‖⋅ D(P). We can obtain a Koksma-Hlawka type inequality by estimating bounds on |f(k)|, where f(k) is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on Walsh coefficients f(k) by introducing an operator called `dyadic difference' ∂i,n. By converting dyadic differences ∂i,n to derivatives (∂ )/(∂ xi), we get a new bound on |f(k)| for a function f whose mixed partial derivatives up to order α in each variable are continuous. This new bound is smaller than the known bound on |f(k)| under some condition. The new Koksma-Hlawka inequality is derived using this new bound on the Walsh coefficients.