Publications

2025

• On the breathing of spectral bands in periodic quantum waveguides with inflating resonators
  
  • Chesnel Lucas
  • Nazarov Sergei A
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2025, 59 (4). We are interested in the lower part of the spectrum of the Dirichlet Laplacian A^ε in a thin waveguide Π^ε obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry Ω by a small factor ε > 0. The Floquet-Bloch theory ensures that the spectrum of A^ε has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to +∞ as O(ε^{-2}) when ε → 0^+. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in Ω that we denote by X_†. Generically, i.e. for most Ω, there holds X_† = {0} and the lower part of the spectrum of A^ε is very sparse, made of bands of length at most O(ε) as ε → 0^+. For certain Ω however, we have dim X_† = 1 and then there are bands of length O(1) which allow for wave propagation in Π^ε. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing Ω around a particular Ω_⋆ where dim X_† = 1. We show a breathing phenomenon for the spectrum of A^ε : when inflating Ω around Ω_⋆ , the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold π^2 /ε^2 , stops breathing and becomes extremely short as Ω continues to inflate.
• Maxwell's equations with hypersingularities at a negative index material conical tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2025, 7 (1), pp.127–169. We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity ε and the permeability µ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in ε and in µ, the corresponding scalar operators are not of Fredholm type in the usual H^1 spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical L^2 framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the T-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.
• Contributions on complexity bounds for Deterministic Partially Observed Markov Decision Process
  
  • Vessaire Cyrille
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Rodríguez-Martínez Alejandro
  Annals of Operations Research, Springer Verlag, 2025, 344 (1), pp.345-382. Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of optimization problems. The Mdp model consists of sets of states, actions, time steps, rewards, and probability transitions. When in a given state and at a given time, the decision maker's action generates a reward and determines the state at the next time step according to the probability transition function. However, Mdps assume that the decision maker knows the state of the controlled dynamical system. Hence, when one needs to optimize controlled dynamical systems under partial observation, one often turns toward the formalism of Partially Observed Markov Decision Processes (Pomdp). Pomdps are often untractable in the general case as Dynamic Programming suffers from the curse of dimensionality. Instead of focusing on the general Pomdps, we present a subclass where transitions and observations mappings are deterministic: Deterministic Partially Observed Markov Decision Processes (Det-Pomdp). That subclass of problems has been studied by (Littman, 1996) and (Bonet, 2009). It was first considered as a limit case of Pomdps by Littman, mainly used to illustrate the complexity of Pomdps when considering as few sources of uncertainties as possible. In this paper, we improve on Littman's complexity bounds. We then introduce and study an even simpler class: Separated Det-Pomdps and give some new complexity bounds for this class. This new class of problems uses a property of the dynamics and observation to push back the curse of dimensionality. (10.1007/s10479-024-06282-0)
  DOI : 10.1007/s10479-024-06282-0
• The non-intrusive reduced basis two-grid method applied to sensitivity analysis
  
  • Grosjean Elise
  • Simeon Bernd
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2025, 59 (1), pp.101-135. This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly ans thus Reduced Basis Methods (RBMs) may be a viable option. We propose new NIRB two-grid algorithms for both the direct and adjoint state methods in the context of parabolic equations. The NIRB two-grid method uses the HF code solely as a “black-box”, requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage employs coarser grids and thus, is significantly less expensive than a fine HF evaluation. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in L∞(0, T ; H10(Ω)), and then establish that these rates are recovered by the NIRB two-grid approximation. These results are supported by numerical simulations. We then propose a new procedure that further reduces the computational costs of the online step while only computing a coarse solution of the state equations. On the adjoint state method, we propose a new algorithm that reduces both the state and adjoint solutions. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system. (10.1051/m2an/2024044)
  DOI : 10.1051/m2an/2024044
• Analysis of the interior transmission problem in an unbounded locally perturbed periodic strip
  
  • Haddar Houssem
  • Jenhani Nouha
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025, 20, pp.305-329. <div><p>We analyze the interior transmission problem in a locally perturbed infinite periodic domain, considering the case where the perturbation intersects the periodic background. An equivalent formulation as coupled quasiperiodic problems is obtained by applying the Floquet-Bloch transform. We perform a discretization with respect to the Floquet-Bloch variable and prove the well-posedness of the semi-discretized problem. We then establish some a priori estimates under regularity assumptions that allow us to prove the convergence of the discrete sequence to the solution of the problem.</p></div> (10.3934/ipi.2025028)
  DOI : 10.3934/ipi.2025028
• Scattering of transient waves by an interface with time-modulated jump conditions
  
  • Michaël Darche
  • Assier Raphaël
  • Guenneau S
  • Lombard Bruno
  • Touboul Marie
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2025, 335, pp.923-951. Time modulation of the physical parameters offers interesting new possibilities for wave control. Examples include amplification of waves, harmonic generation and non-reciprocity, without resorting to non-linear mechanisms. Most of the recent studies focus on the time-modulation of the bulk physical properties. However, as the temporal modulation of these properties is difficult to achieve experimentally, we will concentrate here on the special case of an interface with time-varying jump conditions, which is simpler to implement. This work is focused on wave propagation in a one-dimensional medium containing one modulated interface. Properties of the scattered waves are investigated theoretically: energy balance, generation of harmonics, impedance matching and non-reciprocity. A fourth-order numerical method is also developed to simulate transient scattering. Numerical experiments are conducted to validate the numerical scheme and to illustrate the theoretical findings.
• Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (II) A modal approach
  
  • Cassier Maxence
  • Joly Patrick
  • Martínez Luis Alejandro Rosas
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2025. This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in [Cassier, Joly, Rosas Martínez, Z. Angew. Math. Phys. 74 (2023), 115], we develop a method based on the spectral analysis of the underlying non-self-adjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation. (10.48550/arXiv.2312.12231)
  DOI : 10.48550/arXiv.2312.12231
• Withdrawal of: Solution of the Ovals problem
  
  • Chitour Yacine
  • Denzler Jochen
  • Jean Frédéric
  • Trélat Emmanuel
  , 2025. In the previous version of the preprint, we made a mistake in our proposed solution to the Ovals problem (formulated in [3, 24]). The erroneous claim is that the operator A_T, used in the proof of Lemma 2.4.1, is selfadjoint. But this fact is wrong, as kindly pointed out to us by Matthias Baur, Rupert L. Frank, Larry Read and Timo Weidl, whom we warmly thank. At the moment, unfortunately, this mistake seems fatal to us, so the Ovals conjecture remains open. Nevertheless, since our work contains arguments that may be useful to address the conjecture, we let it available as a preprint, with a warning on Section 2.4.
• Early-Reverberation Imaging Functions for Bounded Elastic Domains
  
  • Ducasse Eric
  • Rodriguez Samuel
  • Bonnet Marc
  Acta Acustica, EDP Sciences, 2025. For the ultrasonic inspection of bounded elastic structures, finite-duration imaging functions are derived in the Fourier-Laplace domain.The signals involved are exponentially windowed, so that early reflections are taken into account more strongly than later ones in the imaging methodology.Applying classical approaches to the general case of anisotropic elasticity, we express the Fréchet derivatives of the relevant data-misfit functional with respect to arbitrary perturbations of the mass density and stiffnesses in terms of forward and adjoint solutions.Their definitions incorporate the exponentially decaying weighting. The proposed finite-duration imaging functions are then defined on that basis.As some areas of the structure are less insonified than others, it is necessary to define normalized imaging functions to compensate for these variations.Our approach in particular aims to overcome the difficulty of dealing with bounded domains containing defects not located in direct line of sight from the transducers and measured signals of long duration.For this initiation work, we demonstate the potential of the proposed method on a two-dimensional test case featuring the imaging of mass and elastic stiffness variations in a region of a bounded isotropic medium that is not directly visible from the transducers. (10.2139/ssrn.4854889)
  DOI : 10.2139/ssrn.4854889
• A note on pliability and the openness of the multiexponential map in Carnot groups
  
  • Jean Frédéric
  • Sigalotti Mario
  • Socionovo Alessandro
  , 2025. In recent years, several notions of non-rigidity of horizontal vectors in Carnot groups have been proposed, motivated, in particular, by the characterization of monotone sets and Whitney extension properties. In this note we compare some of these notions.
• On the constitutive behavior of linear viscoelastic solids under the plane stress condition
  
  • Guzina Bojan B
  • Bonnet Marc
  Journal of Elasticity, Springer Verlag, 2025, 157, pp.45. Motivated by the recent experimental and analytical developments enabling high-fidelity material characterization of (heterogeneous) sheet-like solid specimens, we seek to elucidate the constitutive behavior of linear viscoelastic solids under the plane stress condition. More specifically, our goal is to expose the relationship between the plane-stress viscoelastic constitutive parameters and their (native) ``bulk'' counterparts. To facilitate the sought reduction of the three-dimensional (3D) constitutive behavior, we deploy the concept of projection operators and focus on the frequency-domain behavior by resorting to the Fourier transform and the mathematical framework of tempered distributions, which extends the Fourier analysis to functions (common in linear viscoelasticity) for which Fourier integrals are not convergent. In the analysis, our primary focus is the on class of linear viscoelastic solids whose 3D rheological behavior is described by a set of constant-coefficient ordinary differential equations, each corresponding to a generic arrangement of ``springs'' and ``dashpots''. On reducing the general formulation to the isotropic case, we proceed with an in-depth investigation of viscoelastic solids whose bulk and shear modulus each derive from a suite of classical ``spring and dashpot'' configurations. To enable faithful reconstruction of the 3D constitutive parameters of natural and engineered solids via (i) thin-sheet testing and (ii) applications of the error-in-constitutive-relation approach to the inversion of (kinematic) sensory data, we also examine the reduction of thermodynamic potentials describing linear viscoelasticity under the plane stress condition. The analysis is complemented by a set of analytical and numerical examples, illustrating the effect on the plane stress condition on the behavior of isotropic and anisotropic viscoelastic solids. (10.1007/s10659-025-10136-6)
  DOI : 10.1007/s10659-025-10136-6
• An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 2: transmission problems.
  
  • Cassier Maxence
  • Joly Patrick
  , 2025. In this second chapter, we analyse transmission problems between a dielectric and a dispersive negative material. In the first part, we consider a transmission problem between two half-spaces, filled respectively by the vacuum and a Drude material, and separated by a planar interface. In this setting, we answer the following question: does this medium satisfy a limiting amplitude principle? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. In the second part, we consider the transmission problem of an infinite strip made of a Drude material embedded in the vacuum and analyse the existence and dispersive properties of guided waves. In both problems, our spectral analysis elucidates new and unusual physical phenomena for the considered transmission problems due to the presence of the dispersive negative material. In particular, we prove the existence of an interface resonance in the first part and the existence of slow light phenomena for guiding waves in the second part.
• Efficient boundary integral method to evaluate the acoustic scattering from coupled fluid-fluid problems excited by multiple sources
  
  • Pacaut Louise
  • Chaillat Stéphanie
  • Mercier Jean-François
  • Serre Gilles
  Journal of Computational Physics, Elsevier, 2025, 524, pp.113736. In the naval industry many applications require to study the behavior of a penetrable obstacle embedded in water, notably in presence of a turbulent flow. Such configuration is encountered in particular when noise is scattered by two-phase fluids, e.g., turbulent flows with air bubbles. Fast and efficient numerical methods are required to compute this scattering in the presence of realistic 3D geometries, such as bubble curtains. In [1], we have developed a very efficient approach in the case of a rigid obstacle of arbitrary shape, excited by a turbulent flow. It is based on the numerical evaluation of tailored Green's functions. Here we extend this fast method to the case of a penetrable obstacle. It is not a straightforward extension and we propose two main contributions. First, tailored Green's functions for a fluid-fluid coupled problem are derived theoretically and determined numerically. Second, we show the need of a regularized Boundary Integral formulation to obtain these Green's functions accurately in all configurations. Finally, we illustrate the efficiency of the method on various applications related to the scattering by multiple bubbles. (10.1016/j.jcp.2025.113736)
  DOI : 10.1016/j.jcp.2025.113736
• A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media
  
  • Pescuma Simone
  • Gabard Gwenael
  • Chaumont-Frelet Théophile
  • Modave Axel
  Journal of Computational Physics, Elsevier, 2025, 534, pp.114009. We consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, we extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, we consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes. (10.1016/j.jcp.2025.114009)
  DOI : 10.1016/j.jcp.2025.114009
• Global solution of Quadratic Problems by Interval Methods and Convex Relaxations
  
  • Elloumi Sourour
  • Lambert Amélie
  • Neveu Bertrand
  • Trombettoni Gilles
  Journal of Global Optimization, Springer Verlag, 2025, 91 (2), pp.331–353. Interval branch-and-bound solvers provide reliable algorithms for handling non-convex optimization problems by ensuring the feasibility and the optimality of the computed solutions, i.e. independently from the floating-point rounding errors. Moreover, these solvers deal with a wide variety of mathematical operators. However, these solvers are not dedicated to quadratic optimization and do not exploit nonlinear convex relaxations in their framework. We present an interval branch-andbound method that can efficiently solve quadratic optimization problems. At each node explored by the algorithm, our solver uses a quadratic convex relaxation which is as strong as a semi-definite programming relaxation, and a variable selection strategy dedicated to quadratic problems. The interval features can then propagate efficiently this information for contracting all variable domains. We also propose to make our algorithm rigorous by certifying firstly the convexity of the objective function of our relaxation, and secondly the validity of the lower bound calculated at each node. In the non-rigorous case, our experiments show significant speedups on general integer quadratic instances, and when reliability is required, our first results show that we are able to handle medium-sized instances in a reasonable running time. (10.1007/s10898-024-01370-8)
  DOI : 10.1007/s10898-024-01370-8
• An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 1: general results.
  
  • Cassier Maxence
  • Joly Patrick
  , 2025. In this chapter, we investigate the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity $\varepsilon$ and magnetic permeability $\mu$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which $\varepsilon$ and $\mu$ are rational functions of the frequency. This leads us to analyse the important classes of non-dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.
• Analysis of time-harmonic electromagnetic problems with elliptic material coefficients
  
  • Ciarlet Patrick
  • Modave Axel
  Mathematical Methods in the Applied Sciences, Wiley, 2025. We consider time-harmonic electromagnetic problems with material coefficients represented by elliptic fields, covering a wide range of complex and anisotropic material media. The properties of elliptic fields are analyzed, with particular emphasis on scalar fields and normal tensor fields. Time-harmonic electromagnetic problems with general elliptic material fields are then studied. Well-posedness results for classical variational formulations with different boundary conditions are reviewed, and hypotheses for the coercivity of the corresponding sesquilinear forms are investigated. Finally, the proposed framework is applied to examples of media used in the literature: isotropic lossy media, geometric media, and gyrotropic media. (10.1002/mma.70318)
  DOI : 10.1002/mma.70318