{"title":"On Tikhonov-type regularization with approximated penalty terms","authors":"A. Neubauer","doi":"10.3934/IPI.2021027","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term <inline-formula><tex-math id=\"M1\">\\begin{document}$ P_0 $\\end{document}</tex-math></inline-formula>, we approximate it by a family of penalty terms <inline-formula><tex-math id=\"M2\">\\begin{document}$ ({P_\\beta}) $\\end{document}</tex-math></inline-formula> having nicer properties and analyze what happens as <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\beta\\to 0 $\\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems and Imaging","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/IPI.2021027","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term \begin{document}$ P_0 $\end{document}, we approximate it by a family of penalty terms \begin{document}$ ({P_\beta}) $\end{document} having nicer properties and analyze what happens as \begin{document}$ \beta\to 0 $\end{document}.
We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.
In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term \begin{document}$ P_0 $\end{document}, we approximate it by a family of penalty terms \begin{document}$ ({P_\beta}) $\end{document} having nicer properties and analyze what happens as \begin{document}$ \beta\to 0 $\end{document}.We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.
期刊介绍：
Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing.
This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.
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