Preprint D98/2012
On piecewise constant level-set (PCLS) methods for the identi cation of discontinuous parameters in ill-posed problems
Adriano De Cezaro | Antonio Leitao
Keywords: Inverse Problems; Discontinuous Parameters; Levelset 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 (according to this approach, the levelset function itself is 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 method 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 the inverse potential problem on a 2D-domain, testing the performance of both methods. Our tests demonstrate that the proposed reconstruction methods are capable of solving the ill-posed problem in a stable way, being able to recover complicated inclusions without any a priori geometrical information on the unknown parameter.