{"title":"AN ALTERNATING DIRECTION METHOD OF MULTIPLIERS WITH THE CONDITIONAL GRADIENT TOTAL VARIATION METHOD FOR LINEAR INVERSE PROBLEMS","authors":"A. Bentbib, A. Bouhamidi, K. Kreit","doi":"10.32523/2306-6172-2023-11-2-4-39","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, we study the ill-posed problem using total variation regularization. To solve such a problem, we use an alternating direction method of multipliers to split our problem into two sub-problems. The novelty of our paper is in the use of the conditional gradient total variation method (CGTV) we have recently introduced. The second split- ting sub-problem is solved by transforming the obtained optimization problem to a general Sylvester matrix equation and then an orthogonal projection method is used to solve the obtained matrix equation. We give proof of the convergence of this method. Some numerical examples and applications to image restoration are given to illustrate the effectiveness of the proposed method.","PeriodicalId":42910,"journal":{"name":"Eurasian Journal of Mathematical and Computer Applications","volume":"1 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-01-01","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/10.32523/2306-6172-2023-11-2-4-39","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
Abstract In this paper, we study the ill-posed problem using total variation regularization. To solve such a problem, we use an alternating direction method of multipliers to split our problem into two sub-problems. The novelty of our paper is in the use of the conditional gradient total variation method (CGTV) we have recently introduced. The second split- ting sub-problem is solved by transforming the obtained optimization problem to a general Sylvester matrix equation and then an orthogonal projection method is used to solve the obtained matrix equation. We give proof of the convergence of this method. Some numerical examples and applications to image restoration are given to illustrate the effectiveness of the proposed method.
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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