Optimal algorithms for linear problems with Gaussian measures
type: Item
instance of: scholarly article
title: Optimal algorithms for linear problems with Gaussian measures
zbMATH Open document ID: 0622.65041
DOI: 10.1216/RMJ-1986-16-4-727
published in: Rocky Mountain Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let H be a separable Hilbert space, S be a linear operator on H, \(\mu\) be a Gaussian measure on H with mean 0 and positive definite correlation operator. The following problem is considered: Given linear information Nf about Sf, one wants to construct ''optimal'' algorithms \(\phi\) such that the error E(\(\phi\) (Nf)-Sf) is minimal in the average sense (i.e., \(\int_{F_ 1}E(\phi (Nf)-Sf)\mu (df)\to \min)\) or in the probabilistic sense (i.e., \(\mu\) (\(\{\) f/E(\(\phi\) (Nf)-Sf)\(\leq \epsilon \})\to \sup\). for \(\epsilon >0)\), where E is a very general error functional. The ''uncertainty'' of a given information operator is defined as the error for optimal algorithms using this information operator. Both for the average and for the probabilistic case the following results are proven: For every adaptive information operator, there exists a non-adaptive information operator with at most the same uncertainty, i.e., adaptation does not help. Optimal algorithms can be constructed as (translated) spline algorithms, i.e., as special affine algorithms.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A50
zbMATH DE Number: 4009667
zbMATH Keywords: optimal reduction of uncertainty
zbMATH Keywords: Hilbert space
zbMATH Keywords: Gaussian measure
zbMATH Keywords: correlation operator
zbMATH Keywords: linear information
zbMATH Keywords: adaptive information
zbMATH Keywords: Optimal algorithms
zbMATH Keywords: spline algorithms
author: Grzegorz W. Wasilkowski
MaRDI profile type: Publication
full work available at URL: rmj-1986-16-4-727
OpenAlex ID: W2027915600
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description: scientific article; zbMATH DE number 4009667