On the randomized complexity of Banach space valued integration
type: Item
instance of: scholarly article
title: On the randomized complexity of Banach space valued integration
zbMATH Open document ID: 1310.65026
author: Stefan Heinrich
author: Aicke Hinrichs
published in: Studia Mathematica
publication date: 2014-11-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1312.3290
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46N40
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 6363767
zbMATH Keywords: complexity of integration
zbMATH Keywords: Banach space-valued integration
zbMATH Keywords: minimal error of integration
zbMATH Keywords: randomized algorithms
zbMATH Keywords: multilevel Monte Carlo algorithm
zbMATH Keywords: Banach space of equal norm type
zbMATH Keywords: optimal convergence
MaRDI profile type: Publication
OpenAlex ID: W2963005190
arXiv ID: 1312.3290
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cites work: Tractability of multivariate problems. Volume II: Standard information for functionals.
cites work: On the type and cotype of Banach spaces
DOI: 10.4064/SM223-3-2
review text: The authors study the complexity of Banach space-valued integration in the randomized setting. Let \(Q=[0,\,1]^d\) and let \(C^r(Q,X)\) be the Banach space of all \(r\)-times continuously differentiable functions \(f:\, Q\to X\) where \(X\) is a Banach space. Let \(Sf = \int_Q f(t)\,dt\). Let \(e_n(S,B)\) denote the \(n\)th minimal error of the integration operator \(S\) on the closed unit ball \(B\subset C^r(Q,X)\) in the randomized setting, that is the minimal possible error among all randomized algorithms approximating \(S\) on \(B\) that use at most \(n\) values of the input function \(f\). The authors investigate the relation of the optimal convergence rate to the geometry of \(X\). It is shown that the minimal errors \(e_n(S,B)\) are bounded by \(c\,n^{-r/d -1+1/p}\) for all \(n\in {\mathbb N}\) with some constant \(c>0\) if and only if \(X\) is a Banach space of equal norm type \(p\in [1,\,2]\).
reviewed by: Manfred Tasche
description: scientific article; zbMATH DE number 6363767