About existence and uniqueness of the restricted total least squares estimation
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.
scientific article; zbMATH DE number 569167
scientific article