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

Cezaro, Adriano de; Leitão, Antônio; Tai, Xue-Cheng


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.

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  • IMEF - Artigos publicados em periódicos