Adriano De Cezaro

A. De Cezaro, M. Haltmeier, A. Leitao et al.

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.

Adriano De Cezaro, B. Tomas Johansson

We investigate uniqueness in the inverse problem of reconstructing simultaneously a spacewise conductivity function and a heat source in the parabolic heat equation from the usual conditions of the direct problem and additional information from a supplementary temperature measurement at a given single instant of time. In the multi-dimensional case, we use Carleman estimates for parabolic equations to obtain a uniqueness result. The given data and the solution domain are sufficiently smooth such that the required norms and the derivatives of the conductivity, the source and the solution of the parabolic heat equation exist and are continuous throughout the solution domain. These assumptions can be further relaxed using more involved estimates and techniques but these lengthy details are not included. Instead, in the special case of the one-dimensional heat equation, we give an alternative and rather straightforward proof of uniqueness of the inverse problem, based on integral representations of the solution together with density results for solutions of the corresponding adjoint problem. In this case, the required regularity conditions on the conductivity, source and the solution of the parabolic heat equation are weakened to classes of integrable functions. Keywords: uniqueness; spacewise conductivity and source; final time measurements; heat equation; Carleman estimates.

Adriano De Cezaro, Fabiana Travessini De Cezaro

In this contribution we show sufficient conditions for simultaneous unique identification of unknown spacewise coefficients and heat source in a parabolic partial differential equation given additional final time measurements. Our approach is based on density, in suitable spaces, of the corresponding adjoint problem. A second issue of this paper is the regularization approach. The sequence of approximated solution is obtained by coupling the nonlinear Landweber iteration with iterated Tikhonov regularization. We show that the parameter-to-solution map satisfies sufficient conditions to prove stability and convergence of approximated solutions for the identification problem. We use a unified discrepancy principle as the stopping criteria. Finally, we apply the developed theory in the inverse identification problem of unknown parameters (perfusion coefficient, metabolic heat source) for the identification of tumor regions by thermography.

Adriano De Cezaro

We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem we propose a Tikhonov-type regularization approach coupled with a level set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability of the regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level set method in some interesting inverse problems arising in elliptic PDE models. Keywords: Level Set Methods, Regularization, Ill-Posed Problems, Piecewise Non-Constant Coefficients

Adriano De Cezaro, Fabiana Travessini De Cezaro

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 \textit{a priori} information on the smoothness of the optical coefficients for with (PAT) is particularly a well-suited imaging modality.

Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli

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.