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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Main Author: A. De Cezaro
Format: Article
Language:English
Published: Wiley 2013-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2013/123643
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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.
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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