{"title":"Structure of the least square solutions to overdetermined systems and its applications to practical inverse problems","authors":"","doi":"10.1007/s13160-023-00640-4","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>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 <em>semi-equivalent deformations</em> 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.</p>","PeriodicalId":50264,"journal":{"name":"Japan Journal of Industrial and Applied Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Japan Journal of Industrial and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13160-023-00640-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍：
Japan Journal of Industrial and Applied Mathematics (JJIAM) is intended to provide an international forum for the expression of new ideas, as well as a site for the presentation of original research in various fields of the mathematical sciences. Consequently the most welcome types of articles are those which provide new insights into and methods for mathematical structures of various phenomena in the natural, social and industrial sciences, those which link real-world phenomena and mathematics through modeling and analysis, and those which impact the development of the mathematical sciences. The scope of the journal covers applied mathematical analysis, computational techniques and industrial mathematics.
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