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Papers by Juliette Leblond
Revue Africaine de Recherche en Informatique et Mathématiques Appliquées
The problem we are dealing with is to recover a Robin coefficient (or impedance) from measurement... more The problem we are dealing with is to recover a Robin coefficient (or impedance) from measurements performed on some part of the boundary of a domain, in the framework of nondestructive testing by the means of Electric Impedance Tomography. The impedance can provide information on the location of a corroded area, as well as on the extent of the damage, which has possibly occurred on an unaccessible part of the boundary. Two different identification algorithms are presented and studied: the first one is based on a Kohn and Vogelius cost function, actually an energetic least squares one, which turns the inverse problem into an optimization one ; as for the second, it makes use of the best approximation in Hardy classes, in order to extend the Cauchy data to the unreachable part of the boundary, and then compute the Robin coefficient from these extended data. Special focus is put on the robustness with respect to noise, both from a mathematical and and numerical point of view. Some num...
arXiv (Cornell University), Jan 29, 2014
arXiv (Cornell University), Apr 27, 2022
In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with... more In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev-Golinskii-Kupin and Favorov-Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set.
Trends in Mathematics, 2017
Fields Institute Communications, 2018
Considering a geometry made of three concentric spherical nested layers, each with constant homog... more Considering a geometry made of three concentric spherical nested layers, each with constant homogeneous conductivity, we establish a uniqueness result in inverse conductivity estimation, from partial boundary data in presence of a known source term. We make use of spherical harmonics and linear algebra computations, that also provide us with stability results and a robust reconstruction algorithm. As an application to electroencephalography (EEG), in a spherical 3-layer head model (brain, skull, scalp), we numerically estimate the skull conductivity from available data (electrical potential at electrodes locations on the scalp, vanishing current flux) and given pointwise dipolar sources in the brain.
In this article, we give asymptotic formulas that link the (first and higher order) moments of th... more In this article, we give asymptotic formulas that link the (first and higher order) moments of the magnetic density distribution (also called magnetization) of a compactly supported magnetized sample with the 0-th and 1st order moments of the vertical component of the magnetic field it produces on a square piece of horizontal plane lying above it. Such a framework typically arises in geosciences when one measures the field generated by a small piece of (magnetized) rock with a magnetic microscope. In such a case, the magnetic moments of the sample are usually unknown and one is interested in recovering them from the data measured with the microscope, i.e. the vertical component of the magnetic field on a planar region above the sample. The formulas given in this article provide a practical way of recovering an estimate of the 0-th order moment of the magnetization (usually also called net moment) from simple computations with the measurements. This estimate is asymptotically exact w...
Математический сборник, 2018
we consider the extremal problem of best approximation to some function f in L 2 (I), with I a su... more we consider the extremal problem of best approximation to some function f in L 2 (I), with I a subset of the circle, by the trace of a Hardy function whose modulus is bounded pointwise by some gauge function on the complementary subset.
Sbornik: Mathematics, 2018
Inverse Problems & Imaging, 2019
Advances in Pure and Applied Mathematics, 2011
Journal of Inverse and Ill-posed Problems, 2017
We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply co... more We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary ∂ Ω partialOmega{\partial\Omega}partialOmega . Assuming Dirichlet and Neumann data available on Γ ⊂ ∂ Ω GammasubsetpartialOmega{\Gamma\subset\partial\Omega}GammasubsetpartialOmega to be real-valued functions in W 1 / 2 , 2 ( Γ ) W1/2,2(Gamma){W^{1/2,2}(\Gamma)}W1/2,2(Gamma) and L 2 ( Γ ) L2(Gamma){L^{2}(\Gamma)}L2(Gamma) classes, respectively, we develop a non-iterative method for solving this ill-posed Cauchy problem choosing L 2 L2{L^{2}}L2 bound of the solution on ∂ Ω ∖ Γ partialOmegasetminusGamma{\partial\Omega\setminus\Gamma}partialOmegasetminusGamma as a regularizing parameter. The present complex-analytic approach also naturally allows imposing additional pointwise constraints on the solution which, on practical side, can help incorporating outlying boundary measurements without changing the boundary into a less regular one.
Complex Analysis and Operator Theory, 2015
Numerische Mathematik, 2002
Linear Algebra and its Applications, 2002
We pose and solve an extremal problem in the Hardy class H 2 of the disc, involving the best appr... more We pose and solve an extremal problem in the Hardy class H 2 of the disc, involving the best approximation of a function on a subarc of the circle by a H 2 function, subject to a constraint on its imaginary part on the complementary arc. A constructive algorithm is presented for the computation of such a best approximant, and the method is illustrated by a numerical example. The whole problem is motivated by boundary parameter identification problems arising in non-destructive control.
Journal of Inverse and Ill-posed Problems, 2003
Journal of Fourier Analysis and Applications, 2009
Revue Africaine de Recherche en Informatique et Mathématiques Appliquées
The problem we are dealing with is to recover a Robin coefficient (or impedance) from measurement... more The problem we are dealing with is to recover a Robin coefficient (or impedance) from measurements performed on some part of the boundary of a domain, in the framework of nondestructive testing by the means of Electric Impedance Tomography. The impedance can provide information on the location of a corroded area, as well as on the extent of the damage, which has possibly occurred on an unaccessible part of the boundary. Two different identification algorithms are presented and studied: the first one is based on a Kohn and Vogelius cost function, actually an energetic least squares one, which turns the inverse problem into an optimization one ; as for the second, it makes use of the best approximation in Hardy classes, in order to extend the Cauchy data to the unreachable part of the boundary, and then compute the Robin coefficient from these extended data. Special focus is put on the robustness with respect to noise, both from a mathematical and and numerical point of view. Some num...
arXiv (Cornell University), Jan 29, 2014
arXiv (Cornell University), Apr 27, 2022
In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with... more In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev-Golinskii-Kupin and Favorov-Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set.
Trends in Mathematics, 2017
Fields Institute Communications, 2018
Considering a geometry made of three concentric spherical nested layers, each with constant homog... more Considering a geometry made of three concentric spherical nested layers, each with constant homogeneous conductivity, we establish a uniqueness result in inverse conductivity estimation, from partial boundary data in presence of a known source term. We make use of spherical harmonics and linear algebra computations, that also provide us with stability results and a robust reconstruction algorithm. As an application to electroencephalography (EEG), in a spherical 3-layer head model (brain, skull, scalp), we numerically estimate the skull conductivity from available data (electrical potential at electrodes locations on the scalp, vanishing current flux) and given pointwise dipolar sources in the brain.
In this article, we give asymptotic formulas that link the (first and higher order) moments of th... more In this article, we give asymptotic formulas that link the (first and higher order) moments of the magnetic density distribution (also called magnetization) of a compactly supported magnetized sample with the 0-th and 1st order moments of the vertical component of the magnetic field it produces on a square piece of horizontal plane lying above it. Such a framework typically arises in geosciences when one measures the field generated by a small piece of (magnetized) rock with a magnetic microscope. In such a case, the magnetic moments of the sample are usually unknown and one is interested in recovering them from the data measured with the microscope, i.e. the vertical component of the magnetic field on a planar region above the sample. The formulas given in this article provide a practical way of recovering an estimate of the 0-th order moment of the magnetization (usually also called net moment) from simple computations with the measurements. This estimate is asymptotically exact w...
Математический сборник, 2018
we consider the extremal problem of best approximation to some function f in L 2 (I), with I a su... more we consider the extremal problem of best approximation to some function f in L 2 (I), with I a subset of the circle, by the trace of a Hardy function whose modulus is bounded pointwise by some gauge function on the complementary subset.
Sbornik: Mathematics, 2018
Inverse Problems & Imaging, 2019
Advances in Pure and Applied Mathematics, 2011
Journal of Inverse and Ill-posed Problems, 2017
We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply co... more We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary ∂ Ω partialOmega{\partial\Omega}partialOmega . Assuming Dirichlet and Neumann data available on Γ ⊂ ∂ Ω GammasubsetpartialOmega{\Gamma\subset\partial\Omega}GammasubsetpartialOmega to be real-valued functions in W 1 / 2 , 2 ( Γ ) W1/2,2(Gamma){W^{1/2,2}(\Gamma)}W1/2,2(Gamma) and L 2 ( Γ ) L2(Gamma){L^{2}(\Gamma)}L2(Gamma) classes, respectively, we develop a non-iterative method for solving this ill-posed Cauchy problem choosing L 2 L2{L^{2}}L2 bound of the solution on ∂ Ω ∖ Γ partialOmegasetminusGamma{\partial\Omega\setminus\Gamma}partialOmegasetminusGamma as a regularizing parameter. The present complex-analytic approach also naturally allows imposing additional pointwise constraints on the solution which, on practical side, can help incorporating outlying boundary measurements without changing the boundary into a less regular one.
Complex Analysis and Operator Theory, 2015
Numerische Mathematik, 2002
Linear Algebra and its Applications, 2002
We pose and solve an extremal problem in the Hardy class H 2 of the disc, involving the best appr... more We pose and solve an extremal problem in the Hardy class H 2 of the disc, involving the best approximation of a function on a subarc of the circle by a H 2 function, subject to a constraint on its imaginary part on the complementary arc. A constructive algorithm is presented for the computation of such a best approximant, and the method is illustrated by a numerical example. The whole problem is motivated by boundary parameter identification problems arising in non-destructive control.
Journal of Inverse and Ill-posed Problems, 2003
Journal of Fourier Analysis and Applications, 2009