Structure of the least square solutions to overdetermined systems and its applications to practical inverse problems

In this paper, we study the structure of the least square solutions to overdetermined systems with no solution. In the main theorem, we prove that if an overdetermined system with no solution is deformed into a system of linear equations by the semi-equivalent deformations defined in this paper, the...

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Bibliographic Details
Published inJapan journal of industrial and applied mathematics Vol. 41; no. 2; pp. 945 - 960
Main Authors Hashizume, Kenji, Maruya, Makoto, Ochi, Takayuki, Takabatake, Toshiaki, Takiguchi, Takashi
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 01.05.2024
Springer Nature B.V
Subjects
Applications of Mathematics
Computational Mathematics and Numerical Analysis
Deformation
Inverse problems
Least squares
Linear equations
Mathematical analysis
Mathematics
Mathematics and Statistics
Original Paper
Theorems
Global navigation satellite system
Non-destructive inspection
Primary 65F20; Secondary 34A55
Overdetermined system
Semi-equivalent deformation
Least square solution
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Summary:In this paper, we study the structure of the least square solutions to overdetermined systems with no solution. In the main theorem, we prove that if an overdetermined system with no solution is deformed into a system of linear equations by the semi-equivalent deformations defined in this paper, then an approximate solution to the original overdetermined system with no solution can be given as the unique least square solution to the deformed system of linear equations. We also introduce some applications of our main theorem to practical inverse problems.
ISSN:0916-7005
1868-937X
DOI:10.1007/s13160-023-00640-4