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

• Autor(a) principal: Cezaro, Adriano de
• Data de Publicação: 2009
• Outros Autores: Leitão, Antonio Carlos Gardel, Tai, Xue-Cheng
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da FURG (RI FURG)
• Texto Completo: 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.