On piecewise constant level-set (PCLS) methods for the identication of discontinuous parameters in ill-posed problems
Adriano De Cezaro | Antonio Leitao
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  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 . 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.