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On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients

Authors
Type
Published Article
Journal
Journal of Applied Mathematics
1110-757X
Publisher
Hindawi Limited
Volume
2013
Pages
1–15
Identifiers
DOI: 10.1155/2013/123643
Source
Keywords
  • Applied Mathematics

Abstract

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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