The Core Problem --- Analysis, Properties, and Behaviour

Mgr. Jana Žáková

Doctoral thesis

Abstract:

A wide range of problems arising in real-world applications needs to be solved as linear approximation problems, since they might contain some errors in data. This thesis focuses on solving such problems with the method of the total least squares and the reduction to the so-called core problem within, which is briefly recapitulated in Part I. Although the core problem concept brought important results on solvability of the vector right-hand side problem, it is not completely true for the problem with matrix right-hand side as the core problem within may not have a TLS solution. Therefore, this thesis aims to examine the 'internal structure' of the matrix right-hand side core problems as well as to 'look around' this problem in order to find possible generalizations. In Part II we build general algebraic framework, which enables to interpret the core problem reduction as the orthogonal projection from the set of general approximation problems onto the set of core problems and partially open the question of the core problem (de)composition and (ir)reducibility. Part III extends the core problem theory with three possible generalizations, namely we present the core problem reductions within the linear approximation problem with tensor right-hand side, the bilinear problem with matrix right-hand side and the multilinear problem with tensor right-hand side. The text of this thesis is complemented by copies of the relevant published articles of the applicant.

Abstract:

A wide range of problems arising in real-world applications needs to be solved as linear approximation problems, since they might contain some errors in data. This thesis focuses on solving such problems with the method of the total least squares and the reduction to the so-called core problem within, which is briefly recapitulated in Part I. Although the core problem concept brought important results on solvability of the vector right-hand side problem, it is not completely true for the problem with matrix right-hand side as the core problem within may not have a TLS solution. Therefore, this thesis aims to examine the 'internal structure' of the matrix right-hand side core problems as well as to 'look around' this problem in order to find possible generalizations. In Part II we build general algebraic framework, which enables to interpret the core problem reduction as the orthogonal projection from the set of general approximation problems onto the set of core problems and partially open the question of the core problem (de)composition and (ir)reducibility. Part III extends the core problem theory with three possible generalizations, namely we present the core problem reductions within the linear approximation problem with tensor right-hand side, the bilinear problem with matrix right-hand side and the multilinear problem with tensor right-hand side. The text of this thesis is complemented by copies of the relevant published articles of the applicant.

Keywords

linear approximation problem, total least squa\-res, core problem, core problem reduction, orthogonal transformation, matrix right-hand side, problem (de)composition, irreducible problem, tensor, tensor right-hand side, bilinear problem, multilinear problem

Language used: English

Date on which the thesis was submitted / produced: 13. 9. 2023

Thesis defence

• Supervisor: doc. Ing. Martin Plešinger, Ph.D., doc. Ing. Martin Plešinger, Ph.D.

Technical University of Liberec

Faculty of Science, Humanities and Education

Doctoral programme / field:
Applied Mathematics / Mathematical models and their applications