Cristóbal Meroño

ETSI Navales, Campus Ciudad Universitaria, Madrid.

email: cj.merono at upm.es

Associate Professor (Profesor Contratado Doctor) at Universidad Politécnica de Madrid. I belong to the M^2ASAI research group at UPM. I am also a member of the research team of the project AGAPI.

Research:

My main research interests lie in Inverse Problems for Partial Differential Equations, specially in Inverse Scattering and the Calderón Problem.

Calderón Problem:

The Calderón Problem is the mathematical model of Electrical Impedance Tomography or EIT. This problem consists in the reconstruction of the conductivity inside a conductor from measurements of voltages and electrical currents made at the boundary of the conductor. I am currently interested in the characterization of DtN maps using the notion of Born approximation. One of the main challenges is to rigorously define this mathematical object in the case of ill- posed problems like the Calderón problem. The existence of the Born approximation leads to new local uniqueness, reconstruction and stability results for the inverse problem.

Latest publications

arXiv
Characterization of Dirichlet-to-Neumann maps via the Born approximation (Preprint)

Carlos Castro, Fabricio Macià, Cristóbal Meroño, and Daniel Sánchez-Mendoza

2026 arXiv abs bib

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 \mathbbR^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.
@misc{Characterization_Born, author = {Castro, Carlos and Macià, Fabricio and Meroño, Cristóbal and Sánchez-Mendoza, Daniel}, title = {Characterization of Dirichlet-to-Neumann maps via the Born approximation (Preprint)}, year = {2026}, eprint = {arXiv:2601.11975}, }
arXiv
The spectrum of Dirichlet-to-Neumann maps for radial conductivities (Preprint)

Thierry Daudé, Fabricio Macià, Cristóbal J. Meroño, and François Nicoleau

2025 arXiv abs bib

The problem of characterizing sequences of real numbers arising as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has received considerable attention. In this work, the authors study this question for DtN maps associated with rotation-invariant elliptic operators on the Euclidean ball. They show that the spectrum can be decomposed into a universal term determined by the boundary values of the conductivity and a sequence of Hausdorff moments of an integrable function, called the Born approximation of the conductivity. They further prove that the Born approximation is locally determined near the boundary by the conductivity, yielding a local uniqueness result for the Calderón problem. A Hölder stability result is also established for the map recovering the conductivity from its Born approximation in suitable Sobolev spaces. Finally, the regularity properties of the Born approximation and their dependence on the conductivity are investigated.
@misc{Spectrum_DtN, author = {Daudé, Thierry and Macià, Fabricio and Meroño, Cristóbal J. and Nicoleau, François}, title = {The spectrum of Dirichlet-to-Neumann maps for radial conductivities (Preprint)}, year = {2025}, eprint = {arXiv:2510.22585}, }
Stable factorization of the Calderón problem via the Born approximation

Thierry Daudé, Fabricio Macià, Cristóbal Meroño, and François Nicoleau

J. Lond. Math. Soc. (2), 2026 doi arXiv abs

The Born approximation of a potential in the context of the Calderón inverse problem is an object that can be formally defined in terms of spectral data of the Dirichlet-to-Neumann map of the corresponding Schrödinger operator. In this article, we prove, in the case of radial potentials in the Euclidean ball and any fixed energy, that the Born approximation is well-defined as a compactly supported radial distribution, and that the Calderón problem can be reformulated as recovering a potential from its Born approximation. In addition, we show that the Born approximation depends locally on the potential and captures exactly its singularities, and that the functional that maps the Born approximation to the potential is Hölder continuous. We also prove that the Born approximation converges to the potential in the high-energy limit. Moreover, we give an explicit formula for the Fourier transform of the Born approximation at any fixed energy, and illustrate how it can be used as the basis of an accurate procedure to approximate a potential from its Dirichlet-to-Neumann map.