Cristóbal J. 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 development of effective reconstruction algorithms using techniques from scattering theory like the 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. For recent results in the radial case see this preprint.

Latest publications

arXiv
Stable factorization of the Calderón problem via the Born approximation (Preprint)

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

2024 arXiv abs bib

In this article we prove the existence of the Born approximation in the context of the radial Calderón problem for Schrödinger operators. This is the inverse problem of recovering a radial potential on the unit ball from the knowledge of the Dirichlet-to-Neumann map (DtN map from now on) of the corresponding Schrödinger operator. The Born approximation naturally appears as the linear component of a factorization of the Calderón problem; we show that the non-linear part, obtaining the potential from the Born approximation, enjoys several interesting properties. First, this map is local, in the sense that knowledge of the Born approximation in a neighborhood of the boundary is equivalent to knowledge of the potential in the same neighborhood, and, second, it is Hölder stable. This shows in particular that the ill-posedness of the Calderón problem arises solely from the linear step, which consists in computing the Born approximation from the DtN map by solving a Hausdorff moment problem. Moreover, we present an effective algorithm to compute the potential from the Born approximation and show a result on reconstruction of singularities. Finally, we use the Born approximation to obtain a partial characterization of the set of DtN maps for radial potentials. The proofs of these results do not make use of Complex Geometric Optics solutions or its analogues; they are based on results on inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of A-amplitude introduced by Barry Simon.
@misc{Radial_Born, author = {Daudé, Thierry and Macià, Fabricio and Meroño, Cristóbal J. and Nicoleau, François}, title = {Stable factorization of the Calderón problem via the Born approximation (Preprint)}, year = {2024}, eprint = {arXiv:2402.06321}, }