Fabricio MaciÃ

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.

Fabricio MaciÃ, Shu Nakamura

We analyze a simple one-hidden-layer neural network with ReLU activation functions and fixed biases, with one-dimensional input and output. We study both continuous and discrete versions of the model, and we rigorously prove the convergence of the learning process with the $L^2$ squared loss function and the gradient descent procedure. We also prove the spectral bias property for this learning process. Several conclusions of this analysis are discussed; in particular, regarding the structure and properties that activation functions should possess, as well as the relationships between the spectrum of certain operators and the learning process. Based on this, we also propose an alternative activation function, the full-wave rectified exponential function (FReX), and we discuss the convergence of the gradient descent with this alternative activation function.