Sesión Análisis Numérico y Optimización

Diciembre 12, 15:30 ~ 15:50

On the regularization of an augmented-Lagrangian approach for the identification of discontinuous parameters in ill-posed problems.

AGNELLI, Juan Pablo

We investigate a level-set-type method for solving ill-posed problems with discontinuous coefficients. The goal is to identify the level sets of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. Imposing a suitable constraint, the level-set function itself is forced to be a piecewise constant function. On the other hand, in order to cope with the ill-posedness of the inverse problem, we consider a Tikhonov-type functional and seek for an approximate but stable solution. Therefore, the inverse problem can be stated and treated as an optimization problem with constraints. To solve this optimization problem we proposed an augmented Lagragian approach. Using tools of abstract convexity is possible to show the existence of a zero duality gap and generalized Lagrangian multipliers. Moreover, here we prove that the method is convergent and stable with respect to the noise level in the data. In other words, we prove that the suggested approch is a regularization method, a result that is important from a practical point of view. Additionally, we present an iterative method to compute approximated solutions of the corresponding inverse problem. Some numerical examples applied to the diffuse optical tomography problem demonstrate the viability of the proposed approach.

Autores: AGNELLI, Juan Pablo / Fernández , Damián / De Cezaro, Adriano / Leitão , Antonio .