{"title":"THE MODIFIED TIKHONOV REGULARIZATION METHOD WITH THE SMOOTHED TOTAL VARIATION FOR SOLVING THE LINEAR ILL-POSED PROBLEMS","authors":"V. Vasin, V. Belyaev","doi":"10.32523/2306-6172-2021-9-2-88-100","DOIUrl":null,"url":null,"abstract":"We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.","PeriodicalId":42910,"journal":{"name":"Eurasian Journal of Mathematical and Computer Applications","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Eurasian Journal of Mathematical and Computer Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.32523/2306-6172-2021-9-2-88-100","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.
期刊介绍：
Eurasian Journal of Mathematical and Computer Applications (EJMCA) publishes carefully selected original research papers in all areas of Applied mathematics first of all from Europe and Asia. However papers by mathematicians from other continents are also welcome. From time to time Eurasian Journal of Mathematical and Computer Applications (EJMCA) will also publish survey papers. Eurasian Mathematical Journal publishes 4 issues in a year. A working language of the journal is English. Main topics are: - Mathematical methods and modeling in mechanics, mining, biology, geophysics, electrodynamics, acoustics, industry. - Inverse problems of mathematical physics: theory and computational approaches. - Medical and industry tomography. - Computer applications: distributed information systems, decision-making systems, embedded systems, information security, graphics.
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