Kaïs Ammari, Carlos Castro
In this paper we propose a numerical method to approximate the best decay rate for some dissipative systems that are bounded perturbation of unbounded skew-adjoint operators. We also give some numerical examples and applications to illustrate the efficiency of this approach.
Carlos Castro, Fabricio MaciÃ, Cristóbal Meroño et al.
The problem of identifying the set of Dirichlet-to-Neumann (DtN) maps arising from conductivities on a smooth domain, among operators acting on functions on the boundary, is a challenging issue in the mathematical analysis of the Calderón inverse problem. This question is also relevant in specific applications since, as the inverse problem is ill-posed, numerically reconstructing a conductivity from the knowledge of its DtN map is particularly delicate. In this article, we address this issue by proving that any DtN map arising from a radial conductivity in the unit ball of $\mathbb{R}^d$ admits an exact representation as a linearized DtN map for a uniquely determined integrable function, that we call the Born approximation. This gives a strong necessary condition for an operator to be a DtN map arising from a radial conductivity. In particular, our results are a starting point towards developing a rigorous foundation for numerous linearization-based methods that are commonly used in the numerical solution of the Calderón inverse problem. We also characterize the Born approximation as a solution to a generalized moment problem that is formally well-defined even for non-radial conductivities. We investigate the uniqueness and structure of general non-radial solutions to this moment problem on the unit disk and provide an algorithm to numerically reconstruct the Born approximation in this setting. We provide numerical experiments to test the resolution and robustness of the Born approximation in different situations.
Filipe Condessa, José Bioucas-Dias, Carlos Castro et al.
We introduce a new supervised algorithm for image classification with rejection using multiscale contextual information. Rejection is desired in image-classification applications that require a robust classifier but not the classification of the entire image. The proposed algorithm combines local and multiscale contextual information with rejection, improving the classification performance. As a probabilistic model for classification, we adopt a multinomial logistic regression. The concept of rejection with contextual information is implemented by modeling the classification problem as an energy minimization problem over a graph representing local and multiscale similarities of the image. The rejection is introduced through an energy data term associated with the classification risk and the contextual information through an energy smoothness term associated with the local and multiscale similarities within the image. We illustrate the proposed method on the classification of images of H&E-stained teratoma tissues.
Juan Antonio Barceló, Carlos Castro, Juan Manuel Reyes
We present an iterative algorithm to compute numerical approximations of the potential for the Schrödinger operator from scattering data. Four different types of scattering data are used as follows: fixed energy, fixed incident angle, backscattering and full data. In the case of fixed energy, the algorithm coincides basically with the one recently introduced by Novikov in [Novikov, R. G., "An iterative approach to non-overdetermined inverse scattering at fixed energy", Sbornik: Mathematics 206 (1), 120-134 (2015)], where some estimates are obtained for large energy scattering data. The numerical results that we present here are consistent with these estimates.