Quasi-Monte Carlo methods can be efficient for integration over products of spheres
type: Item
instance of: scholarly article
title: Quasi-Monte Carlo methods can be efficient for integration over products of spheres
zbMATH Open document ID: 1079.65008
author: F. Y. Kuo
author: Ian H. Sloan
published in: Journal of Complexity
publication date: 2005-04-21T00:00:00Z
review text: The paper deals with multivariate quasi-Monte Carlo (QMC) integration of functions defined on the product of \(d\) copies of the unit sphere like, e.~g., \[ \int_{S^2} \cdots \int_{S^2} f(x_1, \ldots, x_d) d \omega(x_1) \cdots d \omega(x_d) \] where \(S^2 \subseteq \mathbb{R}^3\) denotes the usual unit sphere and \(d \omega(x)\) is the surface measure of the unit sphere. Most deterministic numerical methods for such problems have a cost (the number of function evaluations) that grows exponentially with \(d\). An important theoretical question, discussed in the paper, is the possibility to find a function space setting in which the cost, given by the number of function evaluations \(n(\varepsilon,d)\) needed to reduce the initial error by a factor \(\varepsilon\), is bounded independently on \(d\). The authors prove that it is possible, but with arguments that tell us nothing about the construction of appropriate numerical schemes to reach it. The study is similar in spirit to that made earlier for the product of unit cubes. The integrands \(f\) are assumed to belong to some Sobolev space \(H_d\), which is a tensor product of \(d\) reproducing kernel Hilbert spaces with the factors in the tensor product weighted by certain parameters \(\gamma_{d,j}\), where \(j=1, 2, \dots, d\). The QMC tractability and strong tractability is discussed according to the definition that the tractability in the space \(H_d\) means that \[ n(\varepsilon,d) \leq C \varepsilon^{-p} d^q \] holds for some nonnegative constants \(C\), \(p\), and \(q\) independent of \(\varepsilon\) and \(d\), whereas the strong QMC tractability corresponds to \(q=0\). The main result is the proof that for positive and uniformly bounded parameters \(\gamma_{d,j}\) strong QMC tractability holds if and only if \[ \lim\limits_{d \to \infty} \sup \sum\limits_{j=1}^{d} \gamma_{d,j} < \infty \] and the QMC tractability holds if and only if \[ \lim\limits_{d \to \infty} \sup \frac{\sum\limits_{j=1}^{d} \gamma_{d,j}}{…
reviewed by: J. Kaupužs
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 2160148
zbMATH Keywords: quasi-Monte Carlo methods
zbMATH Keywords: products of spheres
zbMATH Keywords: tractability
zbMATH Keywords: worst-case error
zbMATH Keywords: multivariate quasi-Monte Carlo integration
zbMATH Keywords: Sobolev space
zbMATH Keywords: reproducing kernel Hilbert spaces
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: j.jco.2004.07.001
OpenAlex ID: W2075391389
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DOI: 10.1016/J.JCO.2004.07.001
description: scientific article; zbMATH DE number 2160148