arXiv subjects

Jérémi Dardé

Publications and source records attributed to Jérémi Dardé.

14 recordsLinked to original sources

arXiv2026

On the critical time of observability of the multi-dimensional Baouendi-Grushin equation

We investigate the observability properties of the Baouendi-Grushin equation on a tensorized domain $Ω:= \mathcal{B}_R \times \tilde Ω$, where $\mathcal{B}_R$ is the open ball of radius $R$ in dimension $d \ge 2$, and $\tilde Ω$ is a smooth, bounded, open set of arbitrary dimension. Our main result is a precise calculation of the minimal observability time $T^*$, for tensorized observation sets of the form $ω\times \tilde Ω$, with $ω\subset \mathcal{B}_R$ (internal observation), and $Γ\times \tilde Ω$, with $Γ\subset \partial \mathcal{B}_R$ (boundary observation). The main novelty regards the sufficient condition, that is observability of the system when $T>T^*$. This is established by combining refined observability inequalities on the annulus--or the entire boundary--using Carleman estimates, together with a Lebeau-Robbiano strategy to localize the observation sets.

math.OC↗

arXiv2025

Controllability of a Fluid-Structure Interaction System Governed by the Heat and Damped Beam Equations

In this article, we study the null-controllability and observability properties of a bi-dimensio\-nal fluid-structure interaction system, governed by the heat equation coupled with the damped beam equation. To do so, we prove a global Carleman estimate for the coupled system, making explicit the dependence on the damping parameter. In the course of the study, we demonstrate several inequalities for the beam equation alone, valid for different damping parameter regimes.

math.AP↗

arXiv2024

On the penalization by the perimeter in shape optimization applied to Dirichlet inverse obstacle problem

This paper is devoted to the understanding of regularisation process in the shape optimization approach to the so-called Dirichlet inverse obstacle problem for elliptic operators. More precisely, we study two different regularisations of the very classical shape optimization approach consisting in minimizing a mismatched functional. The first one is an implicit regularisation when working in the class of inclusion having a uniform $\varepsilon$-cone property, a natural class in shape optimization. As this regularity is not trivial to guarantee numerically, we discuss the regularisation by perimeter penalization. We show that this second regularisation provides a stability gain in the minimization process.

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arXiv2023

Immersed Boundary Method for the Complete Electrode Model in Electrical Impedance Tomography

We propose an immersed boundary scheme for the numerical resolution of the Complete Electrode Model in Electrical Impedance Tomography, that we use as a main ingredient in the resolution of inverse problems in medical imaging. Such method allows to use a Cartesian mesh without accurate discretization of the boundary, which is useful in situations where the boundary is complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency with two dimensional direct and inverse problems.

math.NA↗

arXiv2022

The Morozov's principle applied to data assimilation problems

This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation.

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arXiv2022

Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets

We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_ωu$ on a rectangular domain. Sharp controllability results already exist when the control domain $ω$ is a vertical strip, or when $q(x) = x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $ω$. In some geometries for $ω$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when $ω$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-\partial_x^2 + ν^2 q(x)^2$ when $\Re(ν)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.

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arXiv2020

Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography

The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for the Neumann-to-Dirichlet map when the conductivity coefficient decays to zero and/or grows to infinity in some given parts of the domain. This allows passing the necessary monotonicity principles to the limiting case. Subsequently, we show how the monotonicity method generalizes to the definite case of reconstructing either perfectly conducting or perfectly insulating inclusions, as well as to the indefinite case where the perturbed conductivity can take any values between, and including, zero and infinity.

math.AP↗

arXiv2020

Critical time for the observability of Kolmogorov-type equations

This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schrödinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.

math.AP↗

arXiv2016

The "exterior approach" applied to the inverse obstacle problem for the heat equation

In this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification.

math.AP↗

arXiv2016

Stability estimates for Navier-Stokes equations and application to inverse problems

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.

math.AP↗

arXiv2016

On the reachable set for the one-dimensional heat equation

The goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x $\in$ (--L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 \textgreater{} L any function which can be extended analytically on the square {x + iy, |x| + |y| $\le$ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| \textless{} L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.

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arXiv2015

Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data.

math.AP↗

arXiv2015

Electrode modelling: The effect of contact impedance

The most realistic model for current-to-voltage measurements of electrical impedance tomography is the complete electrode model which takes into account electrode shapes and contact impedances at the electrode/object interfaces. When contact impedances are small, numerical instability can be avoided by replacing the complete model with the shunt model in which perfect contacts, that is zero contact impedances, are assumed. In the present work we show that using the shunt model causes only a (almost) linear error with respect to the contact impedances in modelling absolute current-to-voltage measurements. Moreover, we note that the electric potentials predicted by the two models exhibit genuinely different Sobolev regularity properties. This, in particular, causes different convergence rates for finite element approximation of the potentials. The theoretical results are backed up by two dimensional numerical experiments.

math.AP↗

arXiv2012

Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and it is based on the Fréchet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.

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