Journal of Applied Mathematics
Volume 2013 (2013), Article ID 123643, 15 pages
Research Article

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

Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Avenida Italia km 8, 96201-900 Rio Grande, RS, Brazil

Received 24 July 2012; Revised 16 October 2012; Accepted 16 October 2012

Academic Editor: Alicia Cordero

Copyright © 2013 A. De Cezaro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.