Publications

2023

• Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach
  
  • Cassier Maxence
  • Joly Patrick
  • Martínez Luis Alejandro Rosas
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2023, 74 (3), pp.115. It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances. (10.1007/s00033-023-01989-9)
  DOI : 10.1007/s00033-023-01989-9
• Renewable-based charging in green ride-sharing
  
  • Perotti Elisabetta
  • Ospina Ana
  • Bianchin Gianluca
  • Simonetto Andrea
  • Dall’anese Emiliano
  Scientific Reports, Nature Publishing Group, 2023, 13 (1), pp.15425. Governments, regulatory bodies, and manufacturers are proposing plans to accelerate the adoption of electric vehicles (EVs), with the goal of reducing the impact of greenhouse gases and pollutants from internal combustion engines on human health and climate change. In this context, the paper considers a scenario where ride-sharing enterprises utilize a 100%-electrified fleet of vehicles, and seeks responses to the following key question: How can renewable-based EV charging be maximized without disrupting the quality of the ride-sharing services? We propose a new mechanism to promote EV charging during hours of high renewable generation, and we introduce the concept of charge request, which is issued by a power utility company. Our mechanism is inspired by a game-theoretic approach where the power utility company proposes incentives and the ride-sharing platform assigns vehicles to both ride and charge requests; the bargaining mechanism leads to prices and EV assignments that are aligned with the notion of Nash equilibria. Numerical results show that it is possible to shift the EV charging during periods of high renewable generation and adapt to intermittent generation while minimizing the impact on the quality of service. The paper also investigates how the users’ willingness to ride-share affects the charging strategy and the quality of service. (10.1038/s41598-023-42042-z)
  DOI : 10.1038/s41598-023-42042-z
• A stochastic volume approach based on tailored Green’s functions for airfoil noise prediction at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  Journal of Sound and Vibration, Elsevier, 2023, 551, pp.117603. The presence of boundary surfaces in a turbulent flow can result in the enhancement of the radiated acoustic field especially for eddies close to any geometrical singularity. At low Mach number, the best suited prediction methods consist in using an acoustic analogy solved with an integral formulation. In the present study, we focus on the Lighthill's wave equation combined with a tailored Green's function and a new semi-analytical model for the turbulence statistics in the space-frequency domain to extend acoustic analogies to geometries of arbitrary shapes. To validate the model predictions for the leading edge noise and the trailing edge noise, a NACA 0012 airfoil at zero angle of attack is considered and predictions are compared to experimental data. The volume integral approach introduced in this study allows us to study the spatial distribution of the noise sources inside the turbulence volume. In addition, the direct noise radiation associated with the turbulent boundary layer is investigated. (10.1016/j.jsv.2023.117603)
  DOI : 10.1016/j.jsv.2023.117603
• Extrapolation-based Prediction-Correction Methods for Time-varying Convex Optimization
  
  • Bastianello Nicola
  • Carli Ruggero
  • Simonetto Andrea
  Signal Processing, Elsevier, 2023, pp.109089. (10.1016/j.sigpro.2023.109089)
  DOI : 10.1016/j.sigpro.2023.109089
• Scaling of Free Subduction on a Sphere
  
  • Ribe Neil M.
  • Chamolly Alexander
  • Gerardi Gianluca
  • Chaillat Stéphanie
  • Li Zhong-Hai
  , 2023. Because Earth's tectonic plates are doubly curved shells, their mechanical behavior during subduction can differ significantly from that of flat plates. We use the boundary-element method (BEM) to study free (gravity-driven) subduction in axisymmetric and 3-D geometry, with a focus on determining the dimensionless parameters that control the dynamics. The axisymmetric model comprises a shell with thickness $h$ and viscosity $\eta_1$ subducting in an isoviscous planet with radius $R_0$ and viscosity $\eta_2$. The angular radius of the trench is $\theta_t$. Scaling analysis based on thin-shell theory reveals two key dimensionless parameters: a `flexural stiffness' $St = (\eta_1/\eta_2)(h/l_b)^3$ that is also relevant for flat plates, and a new `dynamical sphericity number' $\Sigma_D = (l_b/R_0)\cot\theta_t$ that is unique to spherical geometry. Here $l_b$ is the `bending length', or the sum of the lengths of the slab and of the seaward flexural bulge. The definition of $\Sigma_D$ implies that the dynamical effect of sphericity is greater for small plates than for large ones; we call this the `sphericity paradox'. By contrast, the purely geometric effect of sphericity is opposite, i.e. greater for large plates than for small ones. The dynamical and geometrical effects together imply that sphericity significantly influences subduction at all length scales. We confirm the scaling analysis using BEM numerical solutions, which show that the influence of sphericity on the slab sinking speed (up to a few tens of percent) and on the hoop stress (up to a factor of 2-3) are largest for small plates such as the Juan de Fuca, Cocos and Philippine Sea plates. We next study a 3-D model comprising a plate bounded by a ridge and a semicircular trench subducting in a three-layer earth consisting of an upper mantle, a lower mantle and an inviscid core. We examine the linear stability of the shell to longitudinal perturbations corresponding to buckling, and determine a scaling law for the most unstable wavelength that we compare with the observed shapes of northern/western Pacific trenches.
• Optimisation des grands systèmes
  
  • Carpentier Pierre
  , 2023, pp.81. Le cours "Optimisation des grands systèmes" a été donné durant de nombreuses années à l'ENSTA comme enseignement de troisième année dans le parcours dédié à l'approfondissement en optimisation et recherche opérationnelle. Ces notes constituent le support de ce cours et ont pour but de fournir aux étudiants l'essentiel de ce qu'il faut retenir du cours. Le livre "Décomposition-coordination en optimisation déterministe et stochastique", publié en 2017 chez Springer, reprend et détaille l'ensemble des notions du cours, et présente l'extension de ces notions au cas stochastique. L'objectif de ce cours est de présenter les méthodes mathématiques et algorithmes permettant d'optimiser un système dont la taille et l'hétérogénéité sont telles que les méthodes "classiques" de l'optimisation ne peuvent pas être mises en oeuvre. Dans ce cours, on se limite à la présentation des méthodes de décomposition et coordination dans le cadre de l'optimisation convexe différentiable déterministe. Le cours comprend essentiellement deux parties. Dans la première partie, on cherche à introduire les idées de la décomposition/coordination et à développer les interprétations économiques sur un modèle simple, sans se préoccuper outre mesure de généralité ou de rigueur mathématique. Dans la deuxième partie, on développe une théorie générale basée sur le principe du problème auxiliaire, permettant d'une part de lever les restrictions qui paraissaient essentielles dans la première partie, et d'autre part d'étudier dans un cadre unifié la convergence des algorithmes de coordination. Enfin, un exemple correspondant à un réseau de distribution d'eau potable de grande taille est présenté et sert à illustrer l'ensemble des méthodes présentés durant le cours.
• Relaxed-inertial proximal point type algorithms for problems involving strongly quasiconvex functions
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2023. Introduced in the 1970's by Martinet for minimizing convex functions and extended shortly afterwards by Rockafellar towards monotone inclusion problems, the proximal point algorithm turned out to be a viable computational method for solving various classes of optimization problems even beyond the convex framework. In this talk we propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. The method is then extended for equilibrium functions involving strongly quasiconvex functions. Computational results confirm the theoretical advances.
• Diffraction électromagnétique par une couche mince de nanoparticules réparties aléatoirement : développement asymptotique, conditions effectives et simulations.
  
  • Boucart Amandine
  , 2023. Nous considérons le problème de diffraction, en régime harmonique, d’une onde plane électromagnétique par un objet inhomogène recouvert d’une couche très fine de petites particules parfaitement conductrices distribuées aléatoirement. Nous cherchons à quantifier l’effet de cette couche sur le coefficient de réflexion. La taille des particules, leur espacement et l’épaisseur de la couche sont du même ordre mais petites par rapport à la longueur d’onde incidente et les dimensions de l’objet. Deux difficultés apparaissent : (1) Résoudre numériquement les équations de Maxwell dans ce contexte est extrêmement coûteux en terme de taille mémoire et de temps calcul; (2) la répartition des particules n'est pas connue pour un objet donné. Nous allons supposer que c'est une réalisation d'une répartition supposée aléatoire.Pour contourner ces difficultés, nous proposons alors un modèle effectif, à l’aide d’un développement asymptotique multi-échelle de la solution, où la couche de particules est remplacée par une condition aux bords effective, prescrite sur une surface située au-dessus des particules. Les coefficients qui interviennent dans la condition nécessite la résolution de problèmes, dits de cellule, posés un demi-espace recouvert d'une couche de particules, de taille unitaire, réparties aléatoirement.
• Scattering resonances in unbounded transmission problems with sign-changing coefficient
  
  • Carvalho Camille
  • Moitier Zoïs
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2023, 88 (2), pp.215-257. It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ``quasimodes to resonances'' argument from the black-box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided. (10.1093/imamat/hxad005)
  DOI : 10.1093/imamat/hxad005
• Combined field-only boundary integral equations for PEC electromagnetic scattering problem in spherical geometries
  
  • Faria Luiz
  • Pérez-Arancibia Carlos
  • Turc Catalin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023. We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIE. In this paper we use the Combined Field methodology of Burton and Miller within the field-only BIE approach and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are non zero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one -- a property similar to second kind integral equations.
• Analysis of sampling methods for imaging a periodic layer and its defects
  
  • Boukari Yosra
  • Haddar Houssem
  • Jenhani Nouha
  Inverse Problems, IOP Publishing, 2023, 39 (5), pp.055001. We revisit the differential sampling method introduced in [9] for the identification of a periodic domain and some local perturbation. We provide a theoretical justification of the method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet-Bloch variable. The corner stone of the analysis is the justification of the Generalized Linear Sampling Method in this setting for a single Floquet-Bloch mode. (10.1088/1361-6420/acc19a)
  DOI : 10.1088/1361-6420/acc19a
• Relaxed-inertial proximal point algorithms for problems involving strongly quasiconvex functions
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2023.
• Lecture notes on numerical linear algebra
  
  • Bonnet Marc
  , 2023. Course notes, ENSTA Paris (2nd year and M1 level), 2021.
• Fourier representation of the diffusion MRI signal using layer potentials
  
  • Fang Chengran
  • Wassermann Demian
  • Li Jing-Rebecca
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023, 83 (1), pp.99-121. The diffusion magnetic resonance imaging signal arising from biological tissues can be numerically simulated by solving the Bloch-Torrey partial differential equation. Numerical simulations can facilitate the investigation of the relationship between the diffusion MRI signals and cellular structures. With the rapid advance of available computing power, the diffusion MRI community has begun to employ numerical simulations for model formulation and validation, as well as for imaging sequence optimization. Existing simulation frameworks use the finite difference method, the finite element method, or the Matrix Formalism method to solve the Bloch-Torrey partial differential equation. We propose a new method based on the efficient evaluation of layer potentials. In this paper, the mathematical framework and the numerical implementation of the new method are described. We demonstrate the convergence of our method via numerical experiments and analyze the errors linked to various model and simulation parameters. Since our method provides a Fourier-type representation of the diffusion MRI signal, it can potentially facilitate new physical and biological signal interpretations in the future. (10.1137/21M1439572)
  DOI : 10.1137/21M1439572
• Operations Research Approaches for the Satellite Constellation Design Problem
  
  • Mencarelli Luca
  • Floquet Julien
  • Georges Frédéric
  , 2023.
• Neuron modeling, Bloch-Torrey equation, and their application to brain microstructure estimation using diffusion MRI
  
  • Fang Chengran
  , 2023. Non-invasively estimating brain microstructure that consists of a very large number of neurites, somas, and glial cells is essential for future neuroimaging. Diffusion MRI (dMRI) is a promising technique to probe brain microstructural properties below the spatial resolution of MRI scanners. Due to the structural complexity of brain tissue and the intricate diffusion MRI mechanism, in vivo microstructure estimation is challenging.Existing methods typically use simplified geometries, particularly spheres, and sticks, to model neuronal structures and to obtain analytical expressions of intracellular signals. The validity of the assumptions made by these methods remains undetermined. This thesis aims to facilitate simulationdriven brain microstructure estimation by replacing simplified geometries with realistic neuron geometry models and the analytical intracellular signal expressions with diffusion MRI simulations. Combined with accurate neuron geometry models, numerical dMRI simulations can give accurate intracellular signals and seamlessly incorporate effects arising from, for instance, neurite undulation or water exchange between soma and neurites.Despite these advantages, dMRI simulations have not been widely adopted due to the difficulties in constructing realistic numerical phantoms, the high computational cost of dMRI simulations, and the difficulty in approximating the implicit mappings between dMRI signals and microstructure properties. This thesis addresses the above problems by making four contributions. First, we develop a high-performance opensource neuron mesh generator and make publicly available over a thousand realistic cellular meshes.The neuron mesh generator, swc2mesh, can automatically and robustly convert valuable neuron tracing data into realistic neuron meshes. We have carefully designed the generator to maintain a good balance between mesh quality and size. A neuron mesh database, NeuronSet, which contains 1213 simulation-ready cell meshes and their neuroanatomical measurements, was built using the mesh generator. These meshes served as the basis for further research. Second, we increased the computational efficiency of the numerical matrix formalism method by accelerating the eigendecomposition algorithm and exploiting GPU computing. The speed was increased tenfold. With similar accuracy, the optimized numerical matrix formalism is 20 times faster than the FEM method and 65 times faster than a GPU-based Monte Carlo method. By performing simulations on realistic neuron meshes, we investigated the effect of water exchange between somas and neurites, and the relationship between soma size and signals. We then implemented a new simulation method that provides a Fourier-like representation of the dMRI signals. This method was derived theoretically and implemented numerically. We validated the convergence of the method and showed that the error behavior is consistent with our error analysis. Finally, we propose a simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI. By exploiting the powerful modeling and computational capabilities that are mentioned above, we have built a synthetic database containing the dMRI signals and microstructure parameters of 1.4 million artificial brain voxels. We have shown that this database can help approximate the underlying mappings of the dMRI signals to volume and surface fractions using artificial neural networks.
• Scattering in a partially open waveguide: the forward problem
  
  • Bourgeois Laurent
  • Fliss Sonia
  • Fritsch Jean-François
  • Hazard Christophe
  • Recoquillay Arnaud
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2023, 88, pp.102-151. This paper is dedicated to an acoustic scattering problem in a two-dimensional partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a bounded cross-section, while the right part is bounded in the transverse direction by some Perfectly Matched Layers that mimic the situation of an open waveguide, that is with an unbounded cross-section. We prove well-posedness of such scattering problem and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving that such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation. (10.1093/imamat/hxad004)
  DOI : 10.1093/imamat/hxad004
• Analysis of time-harmonic Maxwell impedance problems in anisotropic media
  
  • Chicaud Damien
  • Ciarlet Patrick
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2023, 55 (3), pp.1969-2000. We consider the time-harmonic Maxwell's equations in anisotropic media. The problem to be solved is an approximation of the diffraction problem, or scattering from bounded objects, that is usually set in some exterior domain in $\mathbb{R}^3$. We consider perfectly conducting objects, so the equations are supplemented with a Dirichlet boundary condition on those objects, and we truncate the exterior domain by imposing an impedance condition on an artificial boundary, to model an approximate radiation condition. The resulting problem is then posed in a bounded domain, with Dirichlet and impedance boundary conditions. In this work, we focus on the mathematical meaning of the impedance condition, precisely in which function space it holds. This relies on a careful analysis of the regularity of the traces of electromagnetic fields, which can be derived thanks to the study of the regularity of the solution to second-order surface PDEs. Then, we prove well-posedness of the model, and we determine the a priori regularity of the fields in the domain and on the boundaries, depending on the geometry, the coefficients and the data. Finally, the discretization of the formulations is presented, with an approximation based on edge finite elements. Error estimates are derived, and a benchmark is provided to discuss those estimates. (10.1137/22M1485413)
  DOI : 10.1137/22M1485413
• Differentiability and Regularization of Parametric Convex Value Functions in Stochastic Multistage Optimization
  
  • Le Franc Adrien
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2023. In multistage decision problems, it is often the case that an initial strategic decision (such as investment) is followed by many operational ones (operating the investment). Such initial strategic decision can be seen as a parameter affecting a multistage decision problem. More generally, we study in this paper a standard multistage stochastic optimization problem depending on a parameter. When the parameter is fixed, Stochastic Dynamic Programming provides a way to compute the optimal value of the problem. Thus, the value function depends both on the state (as usual) and on the parameter. Our aim is to investigate on the possibility to efficiently compute gradients of the value function with respect to the parameter, when these objects exist. When nondifferentiable, we propose a regularization method based on the Moreau-Yosida envelope. We present a numerical test case from day-ahead power scheduling.
• Stability estimate for an inverse problem for the time harmonic magnetic schrödinger operator from the near and far field pattern
  
  • Bellassoued Mourad
  • Haddar Houssem
  • Labidi Amal
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2023, 55 (4), pp.2475-2504. We derive conditional stability estimates for inverse scattering problems related to time harmonic magnetic Schrödinger equation. We prove logarithmic type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions. (10.1137/22M1481956)
  DOI : 10.1137/22M1481956
• Quantifying mixing in arbitrary fluid domains: a Padé approximation approach
  
  • Anderson Thomas G
  • Bonnet Marc
  • Veerapaneni Shravan
  Numerical Algorithms, Springer Verlag, 2023, 93, pp.441-458. We consider the model problem of mixing of passive tracers by an incompressible viscous fluid. Addressing questions of optimal control in realistic geometric settings or alternatively the design of fluid-confining geometries that successfully effect mixing requires a meaningful norm in which to quantify mixing that is also suitable for easy and efficient computation (as is needed, e.g., for use in gradient-based optimization methods). We use the physically inspired reasonable surrogate of a negative index Sobolev norm over the complex fluid mixing domain Ω, a task which could be seen as computationally expensive since it requires the computation of an eigenbasis for L2(Ω) by definition. Instead, we compute a representant of the scalar concentration field in an appropriate Sobolev space in order to obtain an equivalent definition of the Sobolev surrogate norm. The representant, in turn, can be computed to high-order accuracy by a Padé approximation to certain fractional pseudo-differential operators, which naturally leads to a sequence of elliptic problems with an inhomogeneity related to snapshots of the time-varying concentration field. Fast and accurate potential theoretic methods are used to efficiently solve these problems, with rapid per-snapshot mix-norm computation made possible by recent advances in numerical methods for volume potentials. We couple the methodology to existing solvers for Stokes and advection equations to obtain a unified framework for simulating and quantifying mixing in arbitrary fluid domains. We provide numerical results demonstrating the convergence of the new approach as the approximation order is increased. (10.1007/s11075-022-01423-7)
  DOI : 10.1007/s11075-022-01423-7
• Robust capacitated Steiner trees and networks with uniform demands
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Poirion Pierre‐louis
  • Ridremont Thomas
  Networks, Wiley, 2023, 82 (1), pp.3-31. We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in the case of terminals with uniform demands. Formally, we are given a graph, capacity, and cost functions on the edges, a root, a subset of vertices called terminals, and a bound k on the number of possible edge failures. We first study the problem where k=1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Secondly, we consider the problem of computing a minimum-cost survivable network, that is, a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a bilevel formulation (with an attacker and a defender), a cutset-based formulation and a flow-based one. We compare the formulations from a theoretical point of view, and we propose algorithms to solve them and compare their efficiency in practice. (10.1002/net.22143)
  DOI : 10.1002/net.22143
• Scattering in a partially open waveguide: the inverse problem
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2023, 17 (2), pp.463-469. In this paper we consider an inverse scattering problem which consists in retrieving obstacles in a partially embedded waveguide in the acoustic case, the measurements being located on the accessible part of the structure. Such accessible part can be considered as a closed waveguide (with a finite cross section), while the embedded part can be considered as an open waveguide (with an infinite cross section). We propose an approximate model of the open waveguide by using Perfectly Matched Layers in order to simplify the resolution of the inverse problem, which is based on a modal formulation of the Linear Sampling Method. Some numerical results show the efficiency of our approach. This paper can be viewed as a continuation of the article [11], which was focused on the forward problem. (10.3934/ipi.2022052)
  DOI : 10.3934/ipi.2022052
• Ultrasonic imaging in highly heterogeneous backgrounds
  
  • Pourahmadian Fatemeh
  • Haddar Houssem
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2023, 479 (2271). This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time t◦ contains (i) a multiply connected set of viscoelastic, anisotropic, and piece-wise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties, and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step t◦, t1, . . ., multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scattering equation. The incident fields affiliated with distinct pairs of obtained wavefronts are analyzed over the stationary and evolving scatterers for a suit of geometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basis for theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features. (10.1098/rspa.2022.0721)
  DOI : 10.1098/rspa.2022.0721
• Asymptotic Expansion of Transmission Eigenvalues for Anisotropic Thin Layers
  
  • Boujlida Hanen
  • Haddar Houssem
  • Khenissi Moez
  Applicable Analysis, Taylor & Francis, 2023, pp.1-22. We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [19]. We also provide formal derivation in more general cases validating the obtained theoretical result. (10.1080/00036811.2023.2187788)
  DOI : 10.1080/00036811.2023.2187788