arXiv subjects

Romina Gaburro

Publications and source records attributed to Romina Gaburro.

At least 19 recordsLinked to original sources

arXiv2026

Stable boundary determination of a complex anisotropic admittivity and its derivatives from a local Neumann-to-Dirichlet map

We study the classical anisotropic Calder\'on problem associated to the elliptic equation $\text{div}(\sigma\nabla u)=0$, where the complex admittivity $\sigma$ is of the form $\sigma=A(\cdot,a(\cdot))$ in a domain $\Omega\subset\mathbb{R}^n$, $n\ge3$. We establish boundary stability estimates for $\sigma$ and its derivatives of arbitrary order from a local Neumann-to-Dirichlet map. Our results extend those of Comm. Partial Differential Equations, 34 (2009) from the real-valued conductivity setting to complex anisotropic admittivities.

math.AP↗

arXiv2026

The local Calder\'on problem and the determination at the boundary of a complex anisotropic admittivity

We address Calder\'on's problem of stably determining the anisotropic complex admittivity $\sigma$ in a domain $\Omega\subset\mathbb{R}^n$, with $n\geq3$, representing a conducting medium, in terms of a Dirichlet-to-Neumann map locally prescribed on a non-empty portion $\Sigma$ of the boundary of $\Omega$, $\partial\Omega$. $\sigma$ is assumed to be of type $\sigma(\cdot)=A(\cdot,a(\cdot))$ in $\Omega$, where the one-parameter family of complex-symmetric matrices $[\lambda^{-1},\:\lambda]\ni t\mapsto A(\cdot,\: t)$ is assumed to be a-priori known and the scalar function $a$ is unknown. We establish Lipschitz and H\"older stability estimates at the boundary for $\sigma$ and its derivatives of arbitrary order on $\Sigma$, respectively, in terms of the local map.

math.AP↗

arXiv2025

A machine learning approach for image classification in synthetic aperture RADAR

We consider the problem in Synthetic Aperture RADAR (SAR) of identifying and classifying objects located on the ground by means of Convolutional Neural Networks (CNNs). Specifically, we adopt a single scattering approximation to classify the shape of the object using both simulated SAR data and reconstructed images from this data, and we compare the success of these approaches. We then identify ice types in real SAR imagery from the satellite Sentinel-1. In both experiments we achieve a promising high classification accuracy ($\geq$75\%). Our results demonstrate the effectiveness of CNNs in using SAR data for both geometric and environmental classification tasks. Our investigation also explores the effect of SAR data acquisition at different antenna heights on our ability to classify objects successfully.

cs.CV↗

arXiv2025

A uniqueness result in the inverse problem for the anisotropic Schrödinger type equation from local measurements

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $σ$ and $q$ of the equation $-\mbox{div}(σ\nabla u)+qu = 0$ from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion $Σ$ of the boundary $\partial Ω$ of a domain $Ω\subset \mathbb{R}^n$, with $n\geq 3$. We assume that $σ$ and $q$ are \textit{a-priori} known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of $Ω$ with curved interfaces. We prove that $σ$ and $q$ can be uniquely determined in $Ω$ from the knowledge of the local map.

math.AP↗

arXiv2025

The local complex Calderón problem. Stability in a layered medium for a special type of anisotropic admittivity

We deal with Calderón's problem in a layered anisotropic medium $Ω\subset\mathbb{R}^n$, $n\geq 3$, with complex anisotropic admittivity $σ=γA$, where $A$ is a known Lipschitz matrix-valued function. We assume that the layers of $Ω$ are fixed and known and that $γ$ is an unknown affine complex-valued function on each layer. We provide Hölder and Lipschitz stability estimates of $σ$ in terms of an ad hoc misfit functional as well as the more classical Dirichlet to Neumann map localised on some open portion $Σ$ of $\partialΩ$, respectively.

math.AP↗

arXiv2025

Electrical Impedance Tomography for Anisotropic Media: a Machine Learning Approach to Classify Inclusions

We consider the problem in Electrical Impedance Tomography (EIT) of identifying one or multiple inclusions in a background-conducting body $Ω\subset\mathbb{R}^2$, from the knowledge of a finite number of electrostatic measurements taken on its boundary $\partialΩ$ and modelled by the Dirichlet-to-Neumann (D-N) matrix. Once the presence of one inclusion in $Ω$ is established, our model, combined with the machine learning techniques of Artificial Neural Networks (ANN) and Support Vector Machines (SVM), may be used to determine the size of the inclusion, the presence of multiple inclusions, and also that of anisotropy within the inclusion(s). Utilising both real and simulated datasets within a 16-electrode setup, we achieve a high rate of inclusion detection and show that two measurements are sufficient to achieve a good level of accuracy when predicting the size of an inclusion. This underscores the substantial potential of integrating machine learning approaches with the more classical analysis of EIT and the inverse inclusion problem to extract critical insights, such as the presence of anisotropy.

math.NA↗

arXiv2024

Singular solutions for complex second order elliptic equations and their application to time-harmonic diffuse optical tomography

We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by Alessandrini in 1990. Our solutions can be applied to the determination of the optical properties of an anisotropic medium in time-harmonic Diffuse Optical Tomography (DOT).

math.AP↗

arXiv2023

Determining an anisotropic conductivity by boundary measurements: stability at the boundary

We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body by means of the so called local Neumann to Dirichlet map on a curved portion $Σ$ of the boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in \cite{Al-dH-G}, we provide a Hölder stability estimate on $Σ$ when the conductivity is a priori known to be a constant matrix near $Σ$.

math.AP↗

arXiv2022

Stability and reconstruction of a special type of anisotropic conductivity in magneto-acoustic tomography with magnetic induction

We consider the issues of stability and reconstruction of the electrical anisotropic conductivity of biological tissues in a domain $Ω\subset\mathbb{R}^3$ by means of the hybrid inverse problem of magneto-acoustic tomography with magnetic induction (MAT-MI). The class of anisotropic conductivities considered here is of type $σ(\cdot)=A(\cdot,γ(\cdot))$ in $Ω$, where $[λ^{-1}, λ]\ni t\mapsto A(\cdot, t)$ is a one-parameter family of matrix-valued functions which are \textit{a-priori} known to be $C^{1,β}$, allowing us to stably reconstruct $γ$ in $Ω$ in terms of an internal functional $F(σ)$. Our results also extend previous results in MAT-MI where $σ(\cdot) = γ(\cdot) D(\cdot)$, with $D$ an \textit{a-priori} known matrix-valued function on $Ω$ to a more general anisotropic structure which depends non-linearly on the scalar function $γ$ to be reconstructed.

math.AP↗

arXiv2022

Stability for the Calderón's problem for a class of anisotropic conductivities via an ad-hoc misfit functional

We address the stability issue in Calderón's problem for a special class of anisotropic conductivities of the form $σ=γA$ in a Lipschitz domain $Ω\subset\mathbb{R}^n$, $n\geq 3$, where $A$ is a known Lipschitz continuous matrix-valued function and $γ$ is the unknown piecewise affine scalar function on a given partition of $Ω$. We define an ad-hoc misfit functional encoding our data and establish stability estimates for this class of anisotropic conductivity in terms of both the misfit functional and the more commonly used local Dirichlet-to-Neumann map.

math.AP↗

arXiv2021

Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $Ω\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $Ω$ is probed with an input field that is modulated with a fixed harmonic frequency $ω=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. Under suitable conditions that include a range of variability for $k$, we prove a result of Hölder stability of the derivatives of the absorption coefficient $μ_a$ of any order at the boundary $\partialΩ$ in terms of the measurements, in the case when the scattering coefficient $μ_s$ is assumed to be known. The stability estimates rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation.

math.AP↗

arXiv2021

Microlocal analysis of borehole seismic data

Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or in another well. Using microlocal analysis, we study possible approximate reconstruction via linearized, filtered backprojection of an isotropic sound speed in the subsurface for three types of data sets. The sources may form a dense array on the surface, or be located along a line on the surface (walkaway geometry) or in another borehole (crosswell). We show that for the dense array, reconstruction is feasible, with no artifacts in the absence of caustics in the background ray geometry, and mild artifacts in the presence of fold caustics in a sense that we define. In contrast, the walkaway and crosswell data sets both give rise to strong, nonremovable artifacts.

math.AP↗

arXiv2020

Inversion of a SIR-based model: a critical analysis about the application to COVID-19 epidemic

Calibration of a SIR (Susceptibles-Infected-Recovered) model with official international data for the COVID-19 pandemics provides a good example of the difficulties inherent the solution of inverse problems. Inverse modeling is set up in a framework of discrete inverse problems, which explicitly considers the role and the relevance of data. Together with a physical vision of the model, the present work addresses numerically the issue of parameters calibration in SIR models, it discusses the uncertainties in the data provided by international authorities, how they influence the reliability of calibrated model parameters and, ultimately, of model predictions.

q-bio.PE↗

arXiv2020

Full Reciprocity-Gap Waveform Inversion in the frequency domain, enabling sparse-source acquisition

The quantitative reconstruction of sub-surface Earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and defines the Full Reciprocity-gap Waveform Inversion (FRgWI ) method. This misfit functional provides additional features compared to the more traditional least-squares approaches with, in particular, that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be arbitrarily chosen. Based on three-dimensional experiments, FRgWI is shown to behave better than Full Waveform Inversion (FWI) in the same context. Then, it allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. On the other hand, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to cross-talk than the least-squares shot-stacking.

physics.geo-ph↗

arXiv2020

Lipschitz stability at the boundary for time-harmonic diffuse optical tomography

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $Ω\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $Ω$ is probed with an input field that is modulated with a fixed harmonic frequency $ω=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. We prove a result of Lipschitz stability of the absorption coefficient $μ_a$ at the boundary $\partialΩ$ in terms of the measurements in the case when the scattering coefficient $μ_s$ is assumed to be known and $k$ belongs to certain intervals depending on some a-priori bounds on $μ_a$, $μ_s$.

math.AP↗

arXiv2019

Microlocal analysis of Doppler Synthetic Aperture Radar

We study the existence and suppression of artifacts for a Doppler-based Synthetic Aperture Radar (DSAR) system. The idealized air- or space-borne system transmits a continuous wave at a fixed frequency and a co-located receiver measures the resulting scattered waves; a windowed Fourier transform then converts the raw data into a function of two variables: slow time and frequency. Under simplifying assumptions, we analyze the linearized forward scattering map and the feasibility of inverting it via filtered backprojection, using techniques of microlocal analysis which robustly describe how sharp features in the target appear in the data. For DSAR with a straight flight path, there is, as with conventional SAR, a left-right ambiguity artifact in the DSAR image, which can be avoided via beam forming to the left or right. For a circular flight path, the artifact has a more complicated structure, but filtering out echoes coming from straight ahead or behind the transceiver, as well as those outside a critical range, allows one to obtain an artifact-free image. Initially derived under a start-stop approximation widely used in range-based SAR, we show that some of these results are robust and hold under a more realistic approximation.

math.AP↗

arXiv2018

Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization

In this paper, we study the performance of Full Waveform Inversion (FWI) from time-harmonic Cauchy data via conditional well-posedness driven iterative regularization. The Cauchy data can be obtained with dual sensors measuring the pressure and the normal velocity. We define a novel misfit functional which, adapted to the Cauchy data, allows the independent location of experimental and computational sources. The conditional well-posedness is obtained for a hierarchy of subspaces in which the inverse problem with partial data is Lipschitz stable. Here, these subspaces yield piecewise linear representations of the wave speed on given domain partitions. Domain partitions can be adaptively obtained through segmentation of the gradient. The domain partitions can be taken as a coarsening of an unstructured tetrahedral mesh associated with a finite element discretization of the Helmholtz equation. We illustrate the effectiveness of the iterative regularization through computational experiments with data in dimension three. In comparison with earlier work, the Cauchy data do not suffer from eigenfrequencies in the configurations.

math.AP↗

arXiv2017

EIT in a layered anisotropic medium

We consider the inverse problem in geophysics of imaging the subsurface of the Earth in cases where a region below the surface is known to be formed by strata of different materials and the depths and thicknesses of the strata and the (possibly anisotropic) conductivity of each of them need to be identified simultaneously. This problem is treated as a special case of the inverse problem of determining a family of nested inclusions in a medium $Ω\subset\mathbb{R}^n$, $n \geq 3$.

math.AP↗