New Trends in Regularization Methods for Inverse Ill-Posed Problems

RUBEN SPIES

Natal / RN

Congreso; ., XXXV Congresso de Matemática Aplicada e Computational, CNMAC 2014; 2014

Sociedade Brasilera de Matemática Aplicada e Computacional, SBMAC

Several generalizations of the traditional Tikhonov-Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. We will show some results on existence, uniqueness and stability of minimizers for arbitrary penalizers in generalized Tikhonov-Phillips functionals. Also, results on the simultaneous use of weighted spatially varying penalizers of L2 and of bounded variation (BV) type will be shown. Open problems will be discussed and results to signal and image restoration problems will be presented.