About existence and uniqueness of the restricted total least squares estimation
type: Item
instance of: scholarly article
title: About existence and uniqueness of the restricted total least squares estimation
zbMATH Open document ID: 0801.65038
DOI: 10.1007/BF01385761
author: Dieter Kuhn
published in: Numerische Mathematik
publication date: 1994-12-04T00:00:00Z
full work available at URL: https://eudml.org/doc/133744
review text: The total least squares (TLS) method for the solution \(a\) of the linear system \(\underline {Xa} = \underline y\), where both data sets \(\underline X\) and \(\underline y\) are subject to error: \(\underline y = y + \delta\) \((y\) is an ``exact'' vector and \(\delta\) is an observation error); \(\underline X = X + \varepsilon\) \((X\) is the unknown exact regressor and \(\varepsilon\) is an observation error). The TLS-method minimizes the functional \(\theta_{\text{TLS}} (X,a) = \lambda \| X - \underline X \|^ 2 + \| Xa -\underline y \|^ 2\) with weighting parameter \(\lambda\). The qualitative behavior of \(\theta_{\text{TLS}}\) by the technique of Lagrangian multipliers is analyzed. The optimization problem includes additional linear inequality constraints for the estimation \(a\) of the form \(Da-b \geq 0\), where \(D\) is a diagonal matrix. It is shown that there exists a unique estimate \(a\) if the weighting parameter \(\lambda\) is chosen sufficiently large. The results are applied to the coefficient identification problem in linear parabolic differential operators.
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 569167
zbMATH Keywords: observation error
zbMATH Keywords: inverse problem
zbMATH Keywords: total least squares method
zbMATH Keywords: Lagrangian multipliers
zbMATH Keywords: coefficient identification
zbMATH Keywords: linear parabolic differential operators
reviewed by: I. Dimov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 569167