Searcharxiv⌕ Search

arXiv subjects

Juan A. Barceló

Publications and source records attributed to Juan A. Barceló.

6 recordsLinked to original sources
arXiv2022

The Born approximation in the three-dimensional Calderón problem

Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators $-Δ+q $. We study the Born approximation of $q$ in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension $d\ge 3$. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map associated to $-Δ+ q$, which is closely related to a particular moment problem. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.

math.AP↗
arXiv2022

The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $B_R \subset \mathbb{R}^3 $ of radius $R$, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $L^{\infty}$ conductivities, and we present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $\partial B_R$. We then investigate how increasing the radius $R$ affects the quality of the Born approximation, and the existence of a scattering limit as $R\to \infty$. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $-Δ+q$, and strong recovery of singularity results are observed in this case.

math.AP↗
arXiv2021

Numerical approximation of the scattering amplitude in elasticity

We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions $d=2$ and $3$. This requires to approximate first the scattering field, for some incident waves, which can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by G. Vainikko in \cite{V} to solve such equations when considering the Lamé operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.

math.NA↗
arXiv2019

Uniqueness for the inverse fixed angle scattering problem

We present a uniqueness result in dimensions $2$ and $3$ for the inverse fixed angle scattering problem associated to the Schrödinger operator $-Δ+q$, where $q$ is a small real valued potential with compact support in the Sobolev space $W^{β,2}$ with $β>0.$ This result improves the known result, due to Stefanov, in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension $4$, but for more regular potentials in $W^{β,2}$ with $β>2/3$.

math.AP↗
arXiv2018

A new convergent algorithm to approximate potentials from fixed angle scattering data

We introduce a new iterative method to recover a real compact supported potential of the Schrödinger operator from their fixed angle scattering data. The method combines a fixed point argument with a suitable approximation of the resolvent of the Schrödinger operator by partial sums associated to its Born series. Convergence is established for potentials with small norm in certain Sobolev spaces. As an application we show some numerical experiments that illustrate this convergence.

math.AP↗