Multi-level Monte Carlo algorithms for infinite-dimensional integration on \(\mathbb R^{\mathbb N}\)
type: Item
instance of: scholarly article
title: Multi-level Monte Carlo algorithms for infinite-dimensional integration on \(\mathbb R^{\mathbb N}\)
zbMATH Open document ID: 1207.65005
author: Fred J. Hickernell
author: Thomas Müller-Gronbach
author: Ben Niu
author: Klaus Ritter
published in: Journal of Complexity
publication date: 2010-08-03T00:00:00Z
review text: Monte Carlo algorithms are successful tool for a numerical approximation of integrals of functions from different functional classes. In the present paper randomized algorithms for numerical integration with respect to a product probability measure on the sequence space \({\mathbb R}^{\mathbb N}\) are studied. Integrands from reproducing kernel Hilbert spaces, whose kernels are superposition of weighted tensor product, are considered. In the introduction the classes of integrands are briefly discussed. The basic idea to consider the infinite-dimensional integration as a limiting case of high-dimensional integration is developed. The use of sequences of coordinate weights , which give the dependence of the functions from their arguments, is applied and connected with the idea of the tractability of the numerical integration. The reproducing kernels of the infinite-dimensional spaces are studied as the limiting case of high-dimensional reproducing kernels. In Section 2 the basic assumptions on the probability measure, the kernels, the weights and the corresponding reproducing kernel Hilbert spaces are given. The ideas for the functions of infinitely many variables and the integration with respect to the product measure are developed. Two examples, where the measure \(\rho\) being the uniform distribution on \([0,1],\) are provided. The first example is the functional space \(W_{2}^{1}([0,1])\) of absolutely continuous functions with square-integrable derivatives, and for the second example the form of the reproducing kernel, which generate the functional space is given. Many lemmas are proved. In Section 3 a cost model for the analysis of infinite-dimensional quadrature problems is presented. Two particular models- for fixed subset sampling and for variable subspace sampling are developed. The randomized algorithms are used for approximation of integrals of integrands of infinitely many arguments. The notions of the worst-case cost of the algorithms for fixed and vari…
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 68W20
Mathematics Subject Classification ID: 68W40
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 5763690
zbMATH Keywords: infinite-dimensional quadrature
zbMATH Keywords: integration on sequence spaces
zbMATH Keywords: multi-level method
zbMATH Keywords: tractability
zbMATH Keywords: fixed subspace sampling
zbMATH Keywords: variable subspace sampling
zbMATH Keywords: minimal error
zbMATH Keywords: worst-case error
zbMATH Keywords: worst-case cost
zbMATH Keywords: Monte Carlo algorithms
zbMATH Keywords: randomized algorithms
zbMATH Keywords: reproducing kernel Hilbert spaces
reviewed by: Vassil Grozdanov
MaRDI profile type: Publication
full work available at URL: j.jco.2010.02.002
OpenAlex ID: W2149958429
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DOI: 10.1016/J.JCO.2010.02.002
description: scientific article; zbMATH DE number 5763690