On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients
We investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikh...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/123643 |
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| _version_ | 1832559029012922368 |
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| author | A. De Cezaro |
| author_facet | A. De Cezaro |
| author_sort | A. De Cezaro |
| collection | DOAJ |
| description | We investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikhonov-type regularization approach coupled with a level-set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models. |
| format | Article |
| id | doaj-art-b933f636e5e04ae69d882304d9fed63e |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-b933f636e5e04ae69d882304d9fed63e2025-02-03T01:31:05ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/123643123643On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant CoefficientsA. De Cezaro0Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Avenida Italia km 8, 96201-900 Rio Grande, RS, BrazilWe investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikhonov-type regularization approach coupled with a level-set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models.http://dx.doi.org/10.1155/2013/123643 |
| spellingShingle | A. De Cezaro On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients Journal of Applied Mathematics |
| title | On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients |
| title_full | On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients |
| title_fullStr | On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients |
| title_full_unstemmed | On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients |
| title_short | On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients |
| title_sort | on a level set method for ill posed problems with piecewise nonconstant coefficients |
| url | http://dx.doi.org/10.1155/2013/123643 |
| work_keys_str_mv | AT adecezaro onalevelsetmethodforillposedproblemswithpiecewisenonconstantcoefficients |