Juliette Leblond - Profile on Academia.edu (original) (raw)

Papers by Juliette Leblond

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data w... more We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best normconstrained approximation of partial L 2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L 2 function on a subset of the circle by the trace of a H 2 function has been considered in . In the present work, we extend such a formulation to the case where the additional interpolation conditions are imposed. We also obtain some new results that can be applied to the original problem: we carry out stability analysis and propose a novel method of evaluation of the approximation and blow-up rates of the solution in terms of a Lagrange parameter leading to a highly-efficient computational algorithm for solving the problem.

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

We study best approximation to a given function, in the least square sense on a subset of the uni... more We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter.

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

We consider the inverse problem in magnetostatics for recovering the moment of a planar magnetiza... more We consider the inverse problem in magnetostatics for recovering the moment of a planar magnetization from measurements of the normal component of the magnetic field at a distance from the support. Such issues arise in studies of magnetic material in general and in paleomagnetism in particular. Assuming the magnetization is a measure with 𝐿 2 -density, we construct linear forms to be applied on the data in order to estimate the moment. These forms are obtained as solutions to certain extremal problems in Sobolev classes of functions, and their computation reduces to solving an elliptic differential-integral equation, for which synthetic numerical experiments are presented.

Advances in Pure and Applied Mathematics, 2011

We present a constructive method for the robust approximation to solutions of some elliptic equat... more We present a constructive method for the robust approximation to solutions of some elliptic equations in a plane domain from incomplete and corrupted boundary data. We state this inverse problem in generalized Hardy spaces of functions satisfying the conjugate Beltrami equation, of which we give some properties, in the Hilbertian framework. The issue is then reworded as a constrained approximation (bounded extremal) problem which is shown to be well-posed. A practical motivation comes from modelling plasma confinement in a tokamak reactor. There, the particular form of the conductivity coefficient leads to Bessel-exponential type families of solutions of which we establish density properties.

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.

Composition Operators on Generalized Hardy Spaces

Complex Analysis and Operator Theory, 2015

Numerische Mathematik, 2002

We investigate consistency properties of rational approximation of prescribed type in the weighte... more We investigate consistency properties of rational approximation of prescribed type in the weighted Hardy space H 2 -(µ) for the exterior of the unit disk, where µ is a positive symmetric measure on the unit circle T. The question of consistency, which is especially significant for gradient algorithms that compute local minima, concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational. In addition to describing some basic properties of the approximation problem, we prove for measures µ having a rational function distribution (weight) with respect to arclength on T, that consistency holds only under rather restricted conditions.

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

Bounded Extremal and Cauchy–Laplace Problems on the Sphere and Shell

Journal of Fourier Analysis and Applications, 2009

Inverse Problems, 2004

We consider the inverse problems of locating pointwise or small size conductivity defaults in a p... more We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?

Inverse Problems, 1999