Open Access
Translator Disclaimer
July 2017 Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration
Takehito Yoshiki
Hiroshima Math. J. 47(2): 155-179 (July 2017). DOI: 10.32917/hmj/1499392824

Abstract

In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function $f\colon[0,1)^s\rightarrow\mathbb{R}$ by a finite point set $\mathcal{P}\subset[0,1)^s$ is the approximation of the integral $I(f):=\int_{[0,1)^s}f(\mathbf{x})\,d\mathbf{x}$ by the average $I_{\mathcal{P}}(f):=\frac{1}{|\mathcal{P}|}\sum_{\mathbf{x} \in \mathcal{P}}f(\mathbf{x})$. We treat a certain class of point sets $\mathcal{P}$ called digital nets. A Koksma-Hlawka type inequality is an inequality providing an upper bound on the integration error $\text{Err}(f;\mathcal{P}):=I(f)-I_{\mathcal{P}}(f)$ of the form $|\text{Err}(f;\mathcal{P})|\le C\cdot \|f\|\cdot D(\mathcal{P})$. We can obtain a Koksma-Hlawka type inequality by estimating bounds on $|\hat{f}(\mathbf{k})|$, where $\hat{f}(\mathbf{k})$ is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on the Walsh coefficients $\hat{f}(\mathbf{k})$ by introducing an operator called ‘dyadic difference’ $\partial_{i,n}$. By converting dyadic differences $\partial_{i,n}$ to derivatives $\frac{\partial }{\partial x_i}$, we get a new bound on $|\hat{f}(\mathbf{k})|$ for a function $f$ whose mixed partial derivatives up to order $\alpha$ in each variable are continuous. This new bound is smaller than the known bound on $|\hat{f}(\mathbf{k})|$ under some instances. The new Koksma-Hlawka type inequality is derived using this new bound on the Walsh coefficients.

Citation

Download Citation

Takehito Yoshiki. "Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration." Hiroshima Math. J. 47 (2) 155 - 179, July 2017. https://doi.org/10.32917/hmj/1499392824

Information

Received: 8 November 2016; Revised: 7 December 2016; Published: July 2017
First available in Project Euclid: 7 July 2017

zbMATH: 06775346
MathSciNet: MR3679888
Digital Object Identifier: 10.32917/hmj/1499392824

Subjects:
Primary: 65D30, 65D32

Rights: Copyright © 2017 Hiroshima University, Mathematics Program

JOURNAL ARTICLE
25 PAGES


SHARE
Vol.47 • No. 2 • July 2017
Back to Top