On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems

Abstract

We investigate level-set-type methods for solving ill-posed problems with discontinuous (piecewise constant) coefficients. 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 first is based on Tikhonov regularization, while the second is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results (Engl H W et al 1996 Mathematics and its Applications (Dordrecht: Kluwer)) are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving the existence of (generalized) Lagrangian multipliers in the sense of (Rockafellar R T and Wets R J-B 1998 Grundlehren der Mathematischen Wissenschaften (Berlin: Springer)). Numerical experiments are performed for a 2D inverse potential problem (Hettlich F and Rundell W 1996 Inverse Problems 12 251–66), 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).