Regularization of inverse ill-posed problems with generalized Tikhonov-Phillips methods.

RUBEN D. SPIES

Santiago de Chile

Seminario; Seminario MATH AmSud; 2013

Delegación Regional de Cooperación para el Cono Sur y Brasil de la embajada de Francia en Chile y la Comisión Nacional de Investigación Científica y Tecnológica CONICYT

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 penalizers of $L^2$ and of bounded variation (BV) type will be shown. Open problems will be discussed and results to image restoration problems will be presented.