We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem related to photoacoustic tomography and a non-linear problem related to the testing of semiconductor devices.

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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 L 2 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).

In this paper, we propose a level set regularization approach combined with a split strategy for the simultaneous identification of piecewise constant diffusion and absorption coefficients from a finite set of optical tomography data (Neumann-to-Dirichlet data). This problem is a high nonlinear inverse problem combining together the exponential and mildly ill-posedness of diffusion and absorption coefficients, respectively. We prove that the parameter-to-measurement map satisfies sufficient conditions (continuity in the L 1 topology) to guarantee regularization properties of the proposed level set approach. On the other hand, numerical tests considering different configurations bring new ideas on how to propose a convergent split strategy for the simultaneous identification of the coefficients. The behavior and performance of the proposed numerical strategy is illustrated with some numerical examples.

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Quantitative Photoacoustic tomography (QPAT) is an emerging medical imaging modality which offers the possibility of combining the high resolution of the acoustic waves and large contrast of optical waves by quantifying the molecular concentration in biological tissue.In this paper, we prove properties of the forward operator that associate optical parameters from measurements of a reconstructed Photoacoustic image. This is often referred to as the optical inverse problem, that is nonlinear and ill-posed. The proved properties of the forward operator provide sufficient conditions to show regularized properties of approximated solutions obtained by Tikhonov-type approaches. The proposed Tikhonov-type approaches analyzed in this contribution are concerned with physical and numerical issues as well as with a priori information on the smoothness of the optical coefficients for with (PAT) is particularly a well-suited imaging modality.

We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice for these quantities by applying a relaxed version of Morozov's discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers.