Publications

2015

• Interface Motion in Random Media
  
  • Bodineau T.
  • Teixeira A.
  Communications in Mathematical Physics, Springer Verlag, 2015, 334 (2), pp.843 - 865. (10.1007/s00220-014-2152-4)
  DOI : 10.1007/s00220-014-2152-4
• Numerical study of a cylinder model of the diffusion MRI signal for neuronal dendrite trees
  
  • van Nguyen Dang
  • Grebenkov Denis S
  • Le Bihan Denis
  • Li Jing-Rebecca
  Journal of Magnetic Resonance, Elsevier, 2015, 252, pp.103-113. (10.1016/j.jmr.2015.01.008)
  DOI : 10.1016/j.jmr.2015.01.008
• Path-dependent equations and viscosity solutions in infinite dimension
  
  • Cosso Andrea
  • Federico Salvatore
  • Gozzi Fausto
  • Rosestolato Mauro
  • Touzi Nizar
  , 2015. Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply to equations which cannot be treated, up to now, with the known theory of viscosity solutions.
• Infinite horizon problems on stratifiable state-constraints sets
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2015, 258 (4), pp.1430–1460. This paper deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the value function has not enough regularity, or can fail to be the unique constrained viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. Here, we consider the case of a set of constraints having a stratified structure. Under this circumstance, the interior of this set may be empty or disconnected, and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints. In such situations, the classical pointing qualification hypothesis are not relevant. The discontinuous value function is then characterized by means of a system of HJB equations on each stratum that composes the state constraints. This result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur. (10.1016/j.jde.2014.11.001)
  DOI : 10.1016/j.jde.2014.11.001
• Optimal Design for Purcell Three-link Swimmer
  
  • Giraldi Laetitia
  • Martinon Pierre
  • Zoppello Marta
  Physical Review, American Physical Society (APS), 2015, 91 (2), pp.023012. In this paper we address the question of the optimal design for the Purcell 3-link swimmer. More precisely we investigate the best link length ratio which maximizes its displacement. The dynamics of the swimmer is expressed as an ODE, using the Resistive Force Theory. Among a set of optimal strategies of deformation (strokes), we provide an asymptotic estimate of the displacement for small deformations, from which we derive the optimal link ratio. Numerical simulations are in good agreement with this theoretical estimate, and also cover larger amplitudes of deformation. Compared with the classical design of the Purcell swimmer, we observe a gain in displacement of roughly 60%.
• Intermittent process analysis with scattering moments
  
  • Muzy Jean-François
  • Bacry Emmanuel
  • Mallat Stéphane
  • Bruna Joan
  Annals of Statistics, Institute of Mathematical Statistics, 2015, 43 (1), pp.323. Scattering moments provide nonparametric models of random processes with stationary increments. They are expected values of random variables computed with a nonexpansive operator, obtained by iteratively applying wavelet transforms and modulus nonlinearities, which preserves the variance. First- and second-order scattering moments are shown to characterize intermittency and self-similarity properties of multiscale processes. Scattering moments of Poisson processes, fractional Brownian motions, Lévy processes and multifractal random walks are shown to have characteristic decay. The Generalized Method of Simulated Moments is applied to scattering moments to estimate data generating models. Numerical applications are shown on financial time-series and on energy dissipation of turbulent flows. (10.1214/14-AOS1276)
  DOI : 10.1214/14-AOS1276
• Optimal control problems on well-structured domains and stratified feedback controls
  
  • Hermosilla Cristopher
  , 2015. The aim of this dissertation is to study some issues in Control Theory of ordinary differential equations. Optimal control problems with tame state-constraints and feedback controls with stratified discontinuities are of special interest. The techniques employed along the manuscript have been chiefly taken from control theory, nonsmooth analysis, variational analysis, tame geometry, convex analysis and differential inclusions theory. The first part of the thesis is devoted to provide general results and definitions required for a good understanding of the entire manuscript. In particular, a strong invariance criterion adapted to manifolds is presented. Moreover, a short insight into manifolds and stratifications is done. The notions of relatively wedged sets is introduced and in addition, some of its properties are stated. The second part is concerned with the characterization of the Value Function of an optimal control problem with state-constraints. Three cases have been taken into account. The first one treats stratifiable state-constraints, that is, sets that can be decomposed into manifolds of different dimensions. The second case is focused on linear systems with convex state-constraints, and the last one considers convex state-constraints as well, but from a penalization point of view. In the latter situation, the dynamics are nonlinear and verify an absorbing property at the boundary. The third part is about discontinuous feedbacks laws whose singularities form a stratified set on the state-space. This type of controls yields to consider stratified discontinuous ordinary differential equations, which motivates an analysis of existence of solutions and robustness with respect to external perturbation for these equations. The construction of a suboptimal continuous feedback from an optimal one is also addressed in this part. The fourth part is dedicated to investigate optimal control problems on networks. The main feature of this contribution is that no controllability assumption around the junctions is imposed. The results can also be extended to generalized notions of networks, where the junction is not a single point but a manifold.
• A Holder-logarithmic stability estimate for an inverse problem in two dimensions
  
  • Santacesaria Matteo
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2015, 23 (1), pp.51–73. The problem of the recovery of a real-valued potential in the two-dimensional Schrodinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Holder type. The paper develops also several estimates for a non-local Riemann-Hilbert problem which could be of independent interest. (10.1515/jiip-2013-0055)
  DOI : 10.1515/jiip-2013-0055
• Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems
  
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Rossi Francesco
  • Sigalotti Mario
  Communications in Mathematical Physics, Springer Verlag, 2015, 333 (3), pp.1225-1239. We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems. (10.1007/s00220-014-2195-6)
  DOI : 10.1007/s00220-014-2195-6
• Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance
  
  • Bonnard Bernard
  • Claeys Mathieu
  • Cots Olivier
  • Martinon Pierre
  Acta Applicandae Mathematicae, Springer Verlag, 2015, 135 (1), pp.pp.5-45. In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. The optimal solution can be found as an extremal, solution of the Maximum Principle and analyzed with the techniques of geometric control. This leads to a numerical investigation based on so-called indirect methods using the HamPath software. The results are then compared with a direct method implemented within the Bocop toolbox. Finally lmi techniques are used to estimate a global optimum. (10.1007/s10440-014-9947-3)
  DOI : 10.1007/s10440-014-9947-3
• Formal Proofs for Nonlinear Optimization
  
  • Magron Victor
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Werner Benjamin
  Journal of Formalized Reasoning, ASDD-AlmaDL, 2015, 8 (15), pp.1-24. We present a formally verified global optimization framework. Given a semialgebraic or transcendental function f and a compact semialgebraic domain K, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of f over K. This method allows to bound in a modular way some of the constituents of f by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.
• Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation
  
  • Bossy Mireille
  • Champagnat Nicolas
  • Leman Helene
  • Maire Sylvain
  • Violeau Laurent
  • Yvinec Mariette
  ESAIM: Proceedings, EDP Sciences, 2015, 48, pp.420-446. The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g.Bossy et al 2009, Mascagni & Simonov 2004}). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized equation on real size biomolecules, that also require efficient computational geometry algorithms. We compare our results with the deterministic solver APBS. In the second part, we prove a new probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte Carlo algorithm is also derived and tested on a simple test case. (10.1051/proc/201448020)
  DOI : 10.1051/proc/201448020
• Equivalence between Exact and Approximate Controllability for Finite-Dimensional Quantum Systems
  
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Rossi Francesco
  • Sigalotti Mario
  , 2015.
• Fine properties of the subdifferential for a class of one-homogeneous functionals
  
  • Chambolle Antonin
  • Goldman Michael
  • Novaga Matteo
  Advances in Calculus of Variation, Walter de Gruyter GmbH, 2015. We collect here some known results on the subdifferential of one-homogeneous functionals, which are anisotropic and nonhomogeneous variants of the total variation and establish a new relationship between Lebesgue points of the calibrating field and regular points of the level lines of the corresponding calibrated function.
• Prediction of Response to Temozolomide in Low-Grade Glioma Patients Based on Tumor Size Dynamics and Genetic Characteristics
  
  • Mazzocco P
  • Barthélémy Célia
  • Kaloshi G
  • Lavielle Marc
  • Ricard D
  • Idbaih A
  • Psimaras D
  • Renard M-A
  • Alentorn A
  • Honnorat J
  • Delattre J-y
  • Ducray F
  • Ribba B
  CPT: Pharmacometrics and Systems Pharmacology, American Society for Clinical Pharmacology and Therapeutics ; International Society of Pharmacometrics, 2015, 4 (12), pp.728–737. Both molecular profiling of tumors and longitudinal tumor size data modeling are relevant strategies to predict cancer patients' response to treatment. Herein we propose a model of tumor growth inhibition integrating a tumor's genetic characteristics (p53 mutation and 1p/19q codeletion) that successfully describes the time course of tumor size in patients with low-grade gliomas treated with first-line temozolomide chemotherapy. The model captures potential tumor progression under chemotherapy by accounting for the emergence of tissue resistance to treatment following prolonged exposure to temozolomide. Using information on individual tumors' genetic characteristics, in addition to early tumor size measurements, the model was able to predict the duration and magnitude of response, especially in those patients in whom repeated assessment of tumor response was obtained during the first 3 months of treatment. Combining longitudinal tumor size quantitative modeling with a tumor''s genetic characterization appears as a promising strategy to personalize treatments in patients with low-grade gliomas. WHAT IS THE CURRENT KNOWLEDGE ON THE TOPIC? þ First-line temozolomide is frequently used to treat low-grade gliomas (LGG), which are slow-growing brain tumors. The duration of response depends on genetic characteristics such as 1p/19q chromosomal codeletion, p53 mutation, and IDH mutations. However, up to now there are no means of predicting, at the individual level, the duration of the response to TMZ and its potential benefit for a given patient. • WHAT QUESTION DID THIS STUDY ADDRESS? þ The present study assessed whether combining longitudinal tumor size quantitative modeling with a tumor's genetic characterization could be an effective means of predicting the response to temozolomide at the individual level in LGG patients. • WHAT THIS STUDY ADDS TO OUR KNOWLEDGE þ For the first time, we developed a model of tumor growth inhibition integrating a tumor's genetic characteristics which successfully describes the time course of tumor size and captures potential tumor progression under chemotherapy in LGG patients treated with first-line temozolomide. The present study shows that using information on individual tumors' genetic characteristics, in addition to early tumor size measurements, it is possible to predict the duration and magnitude of response to temozolomide. • HOW THIS MIGHT CHANGE CLINICAL PHARMACOLOGY AND THERAPEUTICS þ Our model constitutes a rational tool to identify patients most likely to benefit from temozolomide and to optimize in these patients the duration of temozolomide therapy in order to ensure the longest duration of response to treatment. Response evaluation criteria such as RECIST—or RANO for brain tumors—are commonly used to assess response to anticancer treatments in clinical trials. 1,2 They assign a patient's response to one of four categories, ranging from " complete response " to " disease progression. " Yet, criticisms have been raised regarding the use of such categorical criteria in the drug development process, 3,4 and regulatory agencies have promoted the additional analysis of longitudinal tumor size measurements through the use of quantitative modeling. 5 Several mathematical models of tumor growth and response to treatment have been developed for this purpose. 6,7 These analyses have led to the (10.1002/psp4.54)
  DOI : 10.1002/psp4.54
• Monotone numerical schemes and feedback construction for hybrid control systems
  
  • Ferretti Roberto
  • Zidani Hasnaa
  Journal of Optimization Theory and Applications, Springer Verlag, 2015, 165 (2), pp.507-531. Hybrid systems are a general framework which can model a large class of control systems arising whenever a collection of continuous and discrete dynamics are put together in a single model. In this paper, we study the convergence of monotone numerical approximations of value functions associated to control problems governed by hybrid systems. We discuss also the feedback reconstruction and derive a convergence result for the approximate feedback control law. Some numerical examples are given to show the robustness of the monotone approximation schemes. (10.1007/s10957-014-0637-0)
  DOI : 10.1007/s10957-014-0637-0
• Nonlocal Curvature Flows
  
  • Chambolle Antonin
  • Morini Massimiliano
  • Ponsiglione Marcello
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 218 (3), pp.1263. This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. We introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper allows us to establish also new results. (10.1007/s00205-015-0880-z)
  DOI : 10.1007/s00205-015-0880-z
• A conformal mapping algorithm for the Bernoulli free boundary value problem
  
  • Haddar Houssem
  • Kress Rainer
  Mathematical Methods in the Applied Sciences, Wiley, 2015. We propose a new numerical method for the solution of Bernoulli's free boundary value problem for harmonic functions in a doubly connected domain $D$ in $\real^2$ where an unknown free boundary $\Gamma_0$ is determined by prescribed Cauchy data on $\Gamma_0$ in addition to a Dirichlet condition on the known boundary $\Gamma_1$. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar and Kress~\cite{AkKr,HaKr05} for the solution of a related inverse boundary value problem. For this we interpret the free boundary $\Gamma_0$ as the unknown boundary in the inverse problem to construct $\Gamma_0$ from the Dirichlet condition on $\Gamma_0$ and Cauchy data on the known boundary $\Gamma_1$. Our method for the Bernoulli problem iterates on the missing normal derivative on $\Gamma_1$ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet--Neumann boundary value problem in $D$. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. (10.1002/mma.3708)
  DOI : 10.1002/mma.3708
• Avis en réponse à la saisine HCB - dossier NL-2005-23. Paris, le 21 octobre 2015
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie Anne M. A.
  • Bellivier Florence
  • Berny Philippe
  • Bertheau Yves
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Coléno François
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Eychenne Nathalie
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Jestin André
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie V.
  • Lemaire Olivier O.
  • Lereclus Didier
  • Maximilien Rémi
  • Meurs Eliane
  • Moreau de Bellaing Cédric
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Parzy Daniel
  • Regnault-Roger Catherine
  • Renard Michel
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2015, pp.39 p.. Le Haut Conseil des biotechnologies (HCB) a été saisi le 17 juillet 2015 par les autorités compétentes françaises (le ministère de l’Agriculture, de l’Agroalimentaire et de la Forêt) d’une demande d’avis relative au dossier EFSA-GMO-NL-2005-23 de demande d’autorisation de mise sur le marché du maïs génétiquement modifié 59122 pour la culture, l’importation, la transformation, l’alimentation humaine et animale. Ce dossier a été déposé conjointement par les sociétés Pioneer Hi-Bred International et Mycogen Seeds c/o Dow AgroSciences LLC sur le fondement du règlement (CE) n°1829/2003 auprès de l’Autorité européenne de sécurité des aliments via les autorités compétentes néerlandaises, sous la référence EFSA-GMO-NL-2005-23. Par cette saisine, les autorités compétentes françaises consultent le HCB au stade ultime de la préparation au vote des Etats membres à la Commission européenne. Le Comité scientifique (CS)2 du HCB a examiné le dossier en séance du 24 septembre 2015 sous la présidence de Jean-Christophe Pagès. Le présent avis a été adopté par voie électronique le 21 octobre 2015 et publié le 2 décembre 2015.
• A deterministic approximation method in shape optimization under random uncertainties
  
  • Allaire Grégoire
  • Dapogny Charles
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2015, 1, pp.83-143. This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number N of random variables, and using first-or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions. (10.5802/smai-jcm.5)
  DOI : 10.5802/smai-jcm.5
• Mean field games systems of first order
  
  • Cardaliaguet Pierre
  • Graber Philip Jameson
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2015, 21 (3), pp.690–722. We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.
• Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system
  
  • Leman Hélène
  • Meleard Sylvie
  • Mirrahimi Sepideh
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2015, 20 (2), pp.469-493. To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals? We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.
• Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost
  
  • Bokanowski Olivier
  • Picarelli Athena
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2015, 71 (1), pp.125--163. This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach. (10.1007/s00245-014-9255-3)
  DOI : 10.1007/s00245-014-9255-3
• Identification of magnetic deposits in 2-D axisymmetric eddy current models via shape optimization
  
  • Jiang Zixian
  • Haddar Houssem
  • Lechleiter Armin
  • El-Guedri Mabrouka
  Inverse Problems in Science and Engineering, Taylor & Francis, 2015. The non-destructive control of steam generators is an essential task for the safe and failure-free operation of nuclear power plants. Due to magnetite particles in the cooling water of the plants, a frequent source for failures are magnetic deposits in the cooling loop of steam generators. From eddy current signals measured inside a U-tube in the steam generator, we propose and analyze a regularized shape optimization algorithm to identify magnetic deposits outside the U-tube with either known or unknown physical properties. Motivated by the cylindrical geometry of the U-tubes we assume an axisymmetric problem setting, reducing Maxwell's equations to a 2-D elliptic eddy current problem. The feasibility of the proposed algorithms is illustrated via numerical examples demonstrating in particular the stability of the method with respect to noise.
• Analysis of Some Qualitative Methods for Inverse Electromagnetic Scattering Problems
  
  • Haddar Houssem
  , 2015, pp.51. This chapter provides a comprehensive presentation of some qualitative methods associated with inverse 3D electromagnetic scattering problem from inhomogeneous and anisotropic media. We first discuss the problem in the framework of so-called Born approximation, that leads to a linearisation of the inverse problem. We second present and analyze the application of the Linear Sampling Method to the full non linear problem using multistatic data at a given frequency. We especially focus on a generalization of this method based on an exact characterization of the inclusion shape in terms of the available data. We then discuss the closely related interior transmission problem and associated transmission eigenvalues. We complement each chapter with some open challenging questions as well as references for further readings. (10.1007/978-3-319-19306-9)
  DOI : 10.1007/978-3-319-19306-9