arXiv subjects

Elena Beretta

Publications and source records attributed to Elena Beretta.

At least 19 recordsLinked to original sources

arXiv2026

On the Uniqueness of a Nonlinear Discrete Calderón Problem

The discrete Calderón problem aims at recovering the conductivity on the edges of a graph from boundary measurements, encoded by the discrete Dirichlet-to-Neumann (DtN) map. The problem has been intensively studied in the linear case on square lattices since the seminal works of Curtis and Morrow. In this work, we investigate a nonlinear analogue of the discrete Calderón problem for a semilinear second-order elliptic equation on square lattices. We establish three uniqueness results for the conductivity recovery. First, we show that conductivity-dependent corner excitations allow a layer-by-layer reconstruction of the conductivity. Second, we study the linearization of the nonlinear DtN map at an arbitrary background boundary datum and prove that the conductivity and the background potential are uniquely determined by one pair of nonlinear Cauchy data and the linearized DtN map. Third, we show that the linearized data can be replaced by finitely many nonlinear measurements, which uniquely determine the conductivity.

math.AP↗

arXiv2026

Carleman estimates for backward Cahn-Hilliard-reaction-diffusion problems

We study a backward inverse problem for a coupled Cahn-Hilliard-reaction-diffusion system. Our main result is a Carleman estimate for a fourth-order/second-order parabolic system with cross-diffusion terms, which allows us to derive conditional stability estimates for the reconstruction of past states from a single final-time observation: Hölder stability at positive times and logarithmic stability for the initial datum. We then apply the Carleman estimate to a phase-field tumour growth model coupling the tumour volume fraction with a nutrient concentration. In this setting, we obtain backward uniqueness and quantitative stability for the recovery of early tumour states, improving earlier results based on logarithmic convexity, which only yielded uniqueness under an additional smallness assumption on the chemotaxis coefficient. We also discuss how these results support Lipschitz stability on finite-dimensional admissible sets, which is relevant for ensuring convergence of iterative discretisation algorithms.

math.AP↗

arXiv2026

Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation

We study the inverse boundary value problem of detecting a non-uniform conductivity motivated by pacing-guided ablation in cardiac electrophysiology. At the stationary level, the transmembrane potential $u$ in a region $Ω\subset\mathbb{R}^3$ of cardiac tissue satisfies \[ -\nabla\!\cdot(γ\nabla u)+αu^3=0 \quad \text{in }Ω,\qquad γ\nabla u\cdotν=g \quad \text{on }\partialΩ, \] where $γ$ is an anisotropic conductivity tensor and $α$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u|_{\partialΩ}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subsetΩ$ where $γ$ is piecewise constant. We address the inverse problem of determining $γ$ from the Neumann-to-Dirichlet (NtD) map, assuming that $α$ and $D$ are known. To our knowledge, uniqueness in the case of NtD data with anisotropic conductivities in this nonlinear setting has not been analyzed in previous work. Using a first-order linearization around a nontrivial pacing current, we prove uniqueness for $γ$.

math.AP↗

arXiv2025

Lipschitz Stability for Polyhedral Elastic Inclusions from Partial Data

The paper deals with the inverse problem of determining a polyhedral inclusion compactly contained in an elastic body from boundary measurements of traction and displacement taken on an open portion of the boundary. Both the inclusion and the body are made of homogeneous isotropic material. Under suitable assumptions on the geometry of the unknown inclusion, we prove a constructive Lipschitz stability estimate from the local Dirichlet-to-Neumann map.

math.AP↗

arXiv2025

Optimal control of the fidelity coefficient in a Cahn-Hilliard image inpainting model

We consider an inpainting model proposed by A. Bertozzi et al., which is based on a Cahn-Hilliard-type equation. This equation describes the evolution of an order parameter that represents an approximation of the original image occupying a bounded two-dimensional domain. The given image is assumed to be damaged in a fixed subdomain, and the equation is characterised by a linear reaction term. This term is multiplied by the so-called fidelity coefficient, which is a strictly positive bounded function defined in the undamaged region. The idea is that, given an initial image, the order parameter evolves towards the given image, and this process properly diffuses through the boundary of the damaged region, restoring the damaged image, provided that the fidelity coefficient is large enough. Here, we formulate an optimal control problem based on this fact, namely, our cost functional accounts for the magnitude of the fidelity coefficient. Assuming a singular potential to ensure that the order parameter takes its values in between 0 and 1, we first analyse the control-to-state operator and prove the existence of at least one optimal control, establishing the validity of first-order optimality conditions. Then, under suitable assumptions, we demonstrate second-order optimality conditions.

math.OC↗

arXiv2025

Dislocations in a layered elastic medium with applications to fault detection

We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determining the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth's crust is an isotropic, layered medium with Lamé coefficients piecewise Lipschitz on a known partition and that the fault surface is a graph with respect to an arbitrary coordinate system. These results substantially extend those of the authors in {Arch. Ration. Mech. Anal.} {\bf 263} (2020), n. 1, 71--111.

math.AP↗

arXiv2024

The discrete inverse conductivity problem solved by the weights of an interpretable neural network

In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.

math.NA↗

arXiv2024

A shape derivative algorithm for reconstructing elastic dislocations in geophysics

We consider the inverse problem of determining an elastic dislocation that models a seismic fault in the quasi-static regime of aseismic, creeping faults, from displacement measurements made at the surface of Earth. We derive both a distributed as well as a boundary shape derivative that encodes the change in a misfit functional between the measured and the computed surface displacement under infinitesimal movements of the dislocation and infinitesimal changes in the slip vector, which gives the displacement jump across the dislocation. We employ the shape derivative in an iterative reconstruction algorithm. We present some numerical test of the reconstruction algorithm in a simplified 2D setting.

math.AP↗

arXiv2024

Identifying early tumour states in a Cahn-Hilliard-reaction-diffusion model

In this paper, we tackle the problem of reconstructing earlier tumour configurations starting from a single spatial measurement at a later time. We describe the tumour evolution through a diffuse interface model coupling a Cahn-Hilliard-type equation for the tumour phase field to a reaction-diffusion equation for a key nutrient proportion, also accounting for chemotaxis effects. We stress that the ability to reconstruct earlier tumour states is crucial for calibrating the model used to predict the tumour dynamics and also to identify the areas where the tumour initially began to develop. However, backward-in-time inverse problems are well-known to be severely ill-posed, even for linear parabolic equations. Moreover, we also face additional challenges due to the complexity of a non-linear fourth-order parabolic system. Nonetheless, we can establish uniqueness by using logarithmic convexity methods under suitable a priori assumptions. To further address the ill-posedness of the inverse problem, we propose a Tikhonov regularisation approach that approximates the solution through a family of constrained minimisation problems. For such problems, we analytically derive the first-order necessary optimality conditions. Finally, we develop a computationally efficient numerical approximation of the optimisation problems by employing standard $C^0$-conforming first-order finite elements. We conduct numerical experiments on several pertinent test cases and observe that the proposed algorithm consistently meets expectations, delivering accurate reconstructions of the original ground truth.

math.AP↗

arXiv2024

Iterative algorithms for the reconstruction of early states of prostate cancer growth

The development of mathematical models of cancer informed by time-resolved measurements has enabled personalised predictions of tumour growth and treatment response. However, frequent cancer monitoring is rare, and many tumours are treated soon after diagnosis with limited data. To improve the predictive capabilities of cancer models, we investigate the problem of recovering earlier tumour states from a single spatial measurement at a later time. Focusing on prostate cancer, we describe tumour dynamics using a phase-field model coupled with two reaction-diffusion equations for a nutrient and the local prostate-specific antigen. We generate synthetic data using a discretisation based on Isogeometric Analysis. Then, building on our previous analytical work (Beretta et al., SIAP (2024)), we propose an iterative reconstruction algorithm based on the Landweber scheme, showing local convergence with quantitative rates and exploring an adaptive step size that leads to faster reconstruction algorithms. Finally, we run simulations demonstrating high-quality reconstructions even with long time horizons and noisy data.

math.AP↗

arXiv2024

On an inverse problem with applications in cardiac electrophysiology

In this paper, we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the forward problem, we show that perfectly insulating regions (modeling ischemic regions in the cardiac tissue) can be uniquely determined by partial boundary measurements of the potential.

math.AP↗

arXiv2024

Mathematical analysis of a model-constrained inverse problem for the reconstruction of early states of prostate cancer growth

The availability of cancer measurements over time enables the personalised assessment of tumour growth and therapeutic response dynamics. However, many tumours are treated after diagnosis without collecting longitudinal data, and cancer monitoring protocols may include infrequent measurements. To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumour states by using a single spatial tumour dataset and a biomathematical model describing disease dynamics. We focus on prostate cancer, since aggressive cases of this disease are usually treated after diagnosis. We describe tumour dynamics with a phase-field model driven by a generic nutrient ruled by reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, which is a key prostate cancer biomarker. We first improve previous well-posedness results by further showing that the solution operator is continuously Fréchet differentiable. We then analyse the backward inverse problem concerning the reconstruction of earlier tumour states starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate.

math.AP↗

arXiv2023

Perception, performance, and detectability of conversational artificial intelligence across 32 university courses

The emergence of large language models has led to the development of powerful tools such as ChatGPT that can produce text indistinguishable from human-generated work. With the increasing accessibility of such technology, students across the globe may utilize it to help with their school work -- a possibility that has sparked discussions on the integrity of student evaluations in the age of artificial intelligence (AI). To date, it is unclear how such tools perform compared to students on university-level courses. Further, students' perspectives regarding the use of such tools, and educators' perspectives on treating their use as plagiarism, remain unknown. Here, we compare the performance of ChatGPT against students on 32 university-level courses. We also assess the degree to which its use can be detected by two classifiers designed specifically for this purpose. Additionally, we conduct a survey across five countries, as well as a more in-depth survey at the authors' institution, to discern students' and educators' perceptions of ChatGPT's use. We find that ChatGPT's performance is comparable, if not superior, to that of students in many courses. Moreover, current AI-text classifiers cannot reliably detect ChatGPT's use in school work, due to their propensity to classify human-written answers as AI-generated, as well as the ease with which AI-generated text can be edited to evade detection. Finally, we find an emerging consensus among students to use the tool, and among educators to treat this as plagiarism. Our findings offer insights that could guide policy discussions addressing the integration of AI into educational frameworks.

cs.CY↗

arXiv2022

On the reconstruction of cavities in a nonlinear model arising from cardiac electrophysiology

In this paper, we deal with the problem of determining perfectly insulating regions (cavities) from one boundary measurement in a nonlinear elliptic equation arising from cardiac electrophysiology. Based on the results obtained in [9] we propose a new reconstruction algorithm based on $Γ$-convergence. The relevance and applicability of this approach are then shown through several numerical experiments.

math.AP↗

arXiv2022

Identification of cavities and inclusions in linear elasticity with a phase-field approach

In this paper, we deal with the inverse problem of the shape reconstruction of cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement's measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase-field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modeled as inclusions with a very small elasticity tensor.

math.AP↗

arXiv2022

Lipschitz stable determination of polyhedral conductivity inclusions from local boundary measurements

We consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. We prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map extending in a highly nontrivial way the results obtained in [18] and [19] in the two-dimensional case to the three-dimensional setting.

math.AP↗

arXiv2021

A new threshold reveals the uncertainty about the effect of school opening on diffusion of Covid-19

We aim at clarifying the controversy about the effects of school openings or closures on the course of the Covid-19 pandemic. The mathematical analysis of compartmental models with subpopulations shows that the in-school contact rates affects the overall course of the pandemic only above a certain threshold that separates an influence phase from a non-influence one. The threshold, that we calculate via linear approximation in several cases, seems to appear in all contexts, including outbreaks or new strains upsurge, lockdowns, and vaccination campaigns excluding children, albeit with different values. Our theoretical findings are then confirmed by several data-driven studies that have previously identified the phase transition in specific cases. Specific outcomes of this study are: opposite conclusions reached by studies of the same or similar situations might depend on, possibly small, differences in modeling or in parameter estimation from the very noisy Covid-19 data, that result in identifying different phases; it is possible to keep schools open at any stage of the Covid-19 pandemic, but suitably strict rules must be applied at all times or else this becomes highly detrimental to virus containment efforts; as the threshold during vaccination turns out to correspond to the internal transmission rate that would lead to virus extinction if the school population was isolated, the needed strict control can be sustained only for very brief periods; as a result, either schools will have to face a prolonged closure or children need to be vaccinated as well.

physics.soc-ph↗

arXiv2021

Detecting discriminatory risk through data annotation based on Bayesian inferences

Thanks to the increasing growth of computational power and data availability, the research in machine learning has advanced with tremendous rapidity. Nowadays, the majority of automatic decision making systems are based on data. However, it is well known that machine learning systems can present problematic results if they are built on partial or incomplete data. In fact, in recent years several studies have found a convergence of issues related to the ethics and transparency of these systems in the process of data collection and how they are recorded. Although the process of rigorous data collection and analysis is fundamental in the model design, this step is still largely overlooked by the machine learning community. For this reason, we propose a method of data annotation based on Bayesian statistical inference that aims to warn about the risk of discriminatory results of a given data set. In particular, our method aims to deepen knowledge and promote awareness about the sampling practices employed to create the training set, highlighting that the probability of success or failure conditioned to a minority membership is given by the structure of the data available. We empirically test our system on three datasets commonly accessed by the machine learning community and we investigate the risk of racial discrimination.

cs.LG↗