Preprint D104/2013
On piecewise constant level-set (PCLS) methods for the identi cation of discontinuous parameters in ill-posed problems
Adriano De Cezaro | Antonio Leitao | Xue-Cheng Tai
Keywords: Inverse Problems | Ill-posed equations | Level set methods
We investigate level-set type methods for solving ill-posed problems with discontinuous (piecewise constant) coecients. The goal is to identify the level sets as well as the level values of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. The PCLS approach is used here to parametrize the solution of a given operator equation in terms of a L2 level-set function, i.e., the level-set function itself is assumed to be a piecewise constant function. Two distinct methods are proposed for computing stable solutions of the resulting ill-posed problem: The rst one is based on Tikhonov regularization, while the second method is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results [16] are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving existence of (generalized) Lagrangian multipliers in the sense of [35]. Numerical experiments are performed for a 2D inverse potential problem [22], demonstrating the capabilities of both methods for solving this ill-posed problem in a stable way (complicated inclusions are recovered without any a priori geometrical information on the unknown parameter).