Abstract
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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DeCezaro, A., Leitão, A., Tai, XC. (2009). On Level-Set Type Methods for Recovering Piecewise Constant Solutions of Ill-Posed Problems. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_5
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DOI: https://doi.org/10.1007/978-3-642-02256-2_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02255-5
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