On level-set type methods for recovering piecewise constant solutions of ill-posed problems

• Main Author: Cezaro, Adriano de
• Publication Date: 2009
• Other Authors: Leitão, Antônio, Tai, Xue-Cheng
• Format: Article
• Language: eng
• Source: Repositório Institucional da FURG (RI FURG)
• Download full: http://repositorio.furg.br/handle/1/880
• topic: Tikhonov; Level-set type method

We propose a regularization method for solving ill-posed problems, under the assumption that the solutions are piecewise constant functions with unknown level sets and unknown level values. A level set framework is established for the inverse problem and a Tikhonov regularization approach is proposed. Existence of generalized minimizers for the Tikhonov functional is proven. Moreover, we establish convergence and stability results, characterizing our Tikhonov approach as a regularization method. Based on the necessary conditions of optimality for the Tikhonov functional, a level-set type method is derived and implemented numerically for solving an inverse source problem. This allow us to test the quality of the proposed algorithm.