Publications

2026

• State-constrained optimal control on Wasserstein spaces over Riemannian manifolds
  
  • Treumún Ernesto
  • Zidani Hasnaa
  , 2026. We study a state-constrained optimal control problem in Mayer form on the Wasserstein space P 2 (M ) of a complete (possibly non-compact) Riemannian manifold M . The controlled dynamics is given by a nonlocal continuity equation, where the velocity field depends on both the space variable and the evolving probability measure. In the presence of state constraints, the associated value function may fail to be continuous, which prevents a direct characterization through Hamilton-Jacobi-Bellman equations (HJB). Following a level-set approach, we introduce an auxiliary value function defined on an extended space and prove that its zero-sublevel set recovers the epigraph of the original value function. Our main result shows that this auxiliary function is the unique viscosity solution of a suitable HJB equation on P 2 (M ). To prove uniqueness, we develop a comparison principle based on directional differentiability properties of the squared Wasserstein distance. These properties are shown to hold under a geometric assumption on the underlying manifold, satisfied in particular by Cartan-Hadamard manifolds and spaces with sectional curvature bounded from below. This extends previous results obtained in the unconstrained case and in Euclidean or compact settings to the state-constrained framework on general complete Riemannian manifolds.
• Fluid-structure Green's functions via BEM/BEM coupling for flow induced noise in arbitrary elastic geometries
  
  • Pacaut Louise
  • Chaillat Stéphanie
  • Mercier Jean-François
  • Serre Gilles
  , 2026. We address the challenge of efficiently simulating the noise generated by the interaction of a turbulent flow noise with complex elastic structures, a coupled fluid/structure interaction (FSI) problem. Current approaches typically separate vibro-acoustic and hydro-acoustic contributions, limiting the accuracy of hydrodynamic noise predictions. To overcome this limitation, we develop a numerical method for computing a Green's function tailored to the coupled FSI problem, enabling a monolithic prediction of the radiated noise without separating the two components. This approach not only improves the accuracy of hydrodynamic noise simulations but also significantly reduces computational costs. The Green's function is constructed using a novel integral formulation and solved numerically via a coupled fast BEM/ BEM solver.
• Stochastic Optimal Feedforward-Feedback Control for Partially Observable Sensorimotor Systems
  
  • Berret Bastien
  • Jean Frédéric
  , 2026. Robust control of complex engineered and biological systems hinges on the integration of feedforward and feedback mechanisms. This is exemplified in neural motor control, where feedforward muscle co-contraction complements sensory-driven feedback corrections to ensure stable behaviors. However, deriving a general continuous-time framework to determine such optimal control policies for partially observable, stochastic, nonlinear, and high-dimensional systems remains a formidable computational challenge. Here, we introduce a framework that extends neighboring optimal control by enabling the feedforward plan to explicitly account for feedback uncertainties and latencies. Using statistical linearization, we transform the stochastic problem into an approximately equivalent deterministic optimization within a tractable, augmented state space that retains critical nonlinearities, offering both mechanistic interpretability and theoretical guarantees on approximation fidelity. We apply this framework to human neuromechanics, demonstrating that muscle co-contraction emerges as an optimal adaptation to task demands, given the characteristics of our sensorimotor system. Our results provide a computational foundation for neuromotor control and a generalizable tool for the control of nonlinear stochastic systems.
• Asymptotic models for time-domain scattering by small particles
  
  • Savchuk Adrian
  , 2026. In this manuscript, we address the problem of time-dependent wave scattering by multiple small particles of arbitrary shape. To approximate the solution of the associated boundary-value problem, we derive an asymptotic model that achieves a higher convergence rate compared to existing models. Our method relies on a boundary integral formulation, semi-discretized in space using a Galerkin approach, where the main challenge lies in choosing appropriate basis functions for the Galerkin space. We show that using equilibrium densities as basis functions yields a cubic-convergent asymptotic model that ensures a priori stability in the time domain due to the coercivity properties of the single-layer boundary integral operator. Unlike the case of spheres, where entries can be computed explicitly, the generalized method requires double integration, which becomes computationally expensive as the number of particles increases. To address this, we derive a simplified model that preserves the stability and convergence properties of the original formulation, alongside a lower-order but computationally advantageous Born model, providing a rigorous theoretical error analysis for both. When the distance between obstacles decreases with the small parameter, the cubic convergence of the scattered field can no longer be guaranteed. To overcome this limitation, we derive high-order asymptotic models by enriching the basis functions in the Galerkin space to maintain accuracy even for closely clustered particles. Finally, the proposed framework is applied to more complex interactions between a large obstacle and multiple small particles, and is further extended to time-domain electromagnetic scattering by small spheres. Numerical experiments validate the theoretical stability and performance across all proposed models.
• Metamaterials and Fluid Flows
  
  • Avallone Francesco
  • Bosia Federico
  • Chen Yi
  • Colombo Giada
  • Craster Richard
  • de Ponti Jacopo Maria
  • Fabbiane Nicolò
  • Haberman Michael
  • Hussein Mahmoud
  • Hwang Wontae
  • Iemma Umberto
  • Juhl Abigail
  • Kadic Muamer
  • Kotsonis Marios
  • Laude Vincent
  • Marquet Olivier
  • Mery Fabien
  • Michelis Theodoros
  • Nouh Mostafa
  • Ragni Daniele
  • Touboul Marie
  • Wegener Martin
  • Krushynska Anastasiia
  Nature Communications, Nature Publishing Group, 2026. (10.1038/s41467-026-70163-2)
  DOI : 10.1038/s41467-026-70163-2
• Dominance Properties for Fair Electricity Supply Planning in Collective Self-Consumption
  
  • Jorquera-Bravo Natalia
  • Elloumi Sourour
  • Kedad-Sidhoum Safia
  • Plateau Agnès
  , 2026. <div><p>This study addresses the problem of fair electricity supply planning within collective self-consumption communities, focusing on shared distributed energy sources and a common electricity storage system. The objective is to determine an electricity supply plan that ensures a fair allocation of shared resources. We formulate the electricity supply planning problem as a mixed integer linear programming (MILP) model, subsequently reformulated into a linear programming (LP) model thanks to some dominance properties. We then propose a series of fairness measures for the allocation of green electricity and shared economic benefits, including proportional allocation and max-min fairness. We prove that the dominance properties can be extended in most of these fairness models. We conduct numerical experiments based on a real case study, as well as on a set of generated instances. The results illustrate their impact on the use of green electricity produced, resource allocation, and participant costs. They also underscore the trade-offs between achieving fairness and maintaining operational efficiency, thereby offering insights for the fair management of energy resources in self-consumption communities.</p></div>
• On the Low Autocorrelation Binary Sequence Problem
  
  • Elloumi Sourour
  • Palagi Laura
  , 2026. On the Low Autocorrelation Binary Sequence Problem
• Discretization in multilayered media with high contrasts: is it all about the boundaries?
  
  • Carvalho Camille
  • Chaillat Stéphanie
  • Tsogka Chrysoula
  • Cortes Elsie A
  , 2026. Wave propagation in multilayered media with high material contrasts poses significant numerical challenges, as large variations in wavenumbers lead to strong reflections and complex transmission of the incoming wave field. To address these difficulties, we employ a boundary integral formulation thereby avoiding volumetric discretization. In this framework, the accuracy of the numerical solution depends strongly on how the material interfaces are discretized. In this work, we demonstrate that standard meshing strategies based on resolving the maximum wavenumber across the domain become computationally inefficient in multilayered configurations, where high wavenumbers are confined to localized subdomains. Through a systematic study of multilayer transmission problems, we show that no simple discretization rule based on the maximum wavenumber or material contrasts emerges. Instead, the wavenumber of the background (exterior) medium plays a dominant role in determining the optimal boundary resolution. Building on these insights, we propose an adaptive approach that achieves uniform accuracy and efficient computation across multiple layers. Numerical experiments for a range of multilayer configurations demonstrate the scalability and robustness of the proposed approach.
• Htool-DDM: A C++ library for parallel solvers and compressed linear systems.
  
  • Marchand Pierre
  • Tournier Pierre-Henri
  • Jolivet Pierre
  Journal of Open Source Software, Open Journals, 2026, 11 (118), pp.9279. (10.21105/joss.09279)
  DOI : 10.21105/joss.09279
• Automated far-field sound field estimation combining robotized acoustic measurements and the boundary elements method
  
  • Pascal Caroline
  • Marchand Pierre
  • Chapoutot Alexandre
  • Doaré Olivier
  Acta Acustica, EDP Sciences, 2026. The identification and reconstruction of acoustic fields radiated by unknown structures is usually performed using either Sound Field Estimation or Near-field Acoustic Holography techniques. The latter turns out to be especially useful when data is only available close to the source, but information throughout the whole space is needed. Yet, the lack of amendable and efficient implementations of state-of-the-art solutions, as well as the laborious and often lengthy deployment of acoustic measurements continue to be significant obstacles to the practical application of such methods. The purpose of this work is to address both problems. First, a completely automated metrology setup is proposed, in which a robotic arm is used to gather extensive and accurately positioned acoustic data without any human intervention. The impact of the robot on acoustic pressure measurements is cautiously evaluated, and proved to remain limited below 1 kHz. The Sound Field Estimation is then tackled using the Boundary Element Method, and implemented using the FreeFEM software. Numerically simulated measurements have allowed us to assess the method accuracy, which matches theoretically expected results and proves to remain robust against positioning inaccuracies, provided that the robot is carefully calibrated. The overall solution has been successfully tested using actual robotized measurements of an unknown loudspeaker, with a reconstruction error of less than 30 %. (10.1051/aacus/2026017)
  DOI : 10.1051/aacus/2026017
• Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach
  
  • Baudet Cédric
  Asymptotic Analysis, IOS Press, 2026. We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted ε. We propose in this work an asymptotic expansion of the solution with respect to ε at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates. (10.1177/09217134251389983)
  DOI : 10.1177/09217134251389983
• Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond
  
  • Laplace Mermoud Dylan
  • Simonetto Andrea
  • Elloumi Sourour
  Quantum, Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2026, 10, pp.1998. We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the cx gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, QuPer, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with 256 variables, requiring 20 qubits. (10.22331/q-2026-02-09-1998)
  DOI : 10.22331/q-2026-02-09-1998
• Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  , 2026. <div><p>We study the one-dimensional Helmholtz equation with (possibly perturbed) quasiperiodic coefficients. Quasiperiodic functions are the restriction of higher dimensional periodic functions along a certain (irrational) direction. In classical settings, for real-valued frequencies, this equation is generally not well-posed: existence of solutions in L 2 is not guaranteed and uniqueness in L ∞ may fail. This is a well-known difficulty of Helmholtz equations, but it has never been addressed in the quasiperiodic case. We tackle this issue by using the limiting absorption principle, which consists in adding some imaginary part (also called absorption) to the frequency in order to make the equation well-posed in L 2 , and then defining the physically relevant solution by making the absorption tend to zero. In previous work, we introduced a definition of the solution of the equation with absorption based on Dirichlet-to-Neumann (DtN) boundary conditions. This approach offers two key advantages: it facilitates the limiting process and has a direct numerical counterpart. In this work, we first explain why the DtN boundary conditions have to be replaced by Robin-to-Robin boundary conditions to make the absorption go to zero. We then prove, under technical assumptions on the frequency, that the limiting absorption principle holds and we propose a numerical method to compute the physical solution.</p></div>
• A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure
  
  • Zhou Yuyuan
  • Audibert Lorenzo
  • Meng Shixu
  • Zhang Bo
  , 2026. In this work we propose a theoretical and computational framework for solving the three dimensional inverse medium scattering problem, based on a set of data-driven basis arising from the linearized problem. This set of data-driven basis consists of generalizations of prolate spheroidal wave functions to three dimensions (3D PSWFs), the main ingredients to explore a low-rank approximation of the inverse solution. We first establish the fundamentals of the inverse scattering analysis, including regularity in a customized Sobolev space and new a priori estimate. This is followed by a computational framework showcasing computing the 3D PSWFs and the low-rank approximation of the inverse solution. These results rely heavily on the fact that the 3D PSWFs are eigenfunctions of both a restricted Fourier integral operator and a Sturm-Liouville differential operator. Furthermore we propose a Tikhonov regularization method with a customized penalty norm and a localized imaging technique to image a targeting object despite the possible presence of its surroundings. Finally various numerical examples are provided to demonstrate the potential of the proposed method.
• Discrete FEM-BEM coupling with the Generalized Optimized Schwarz Method
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2026. The present contribution aims at developing a non-overlapping Domain Decomposition (DD) approach to the solution of acoustic wave propagation boundary value problems based on the Helmholtz equation, on both bounded and unbounded domains. This DD solver, called Generalized Optimized Schwarz Method (GOSM), is a substructuring method, that is, the unknowns of an iteration are associated with the subdomains interfaces. We extend the analysis presented in a previous paper of one of the author to a fully discrete setting. We do not consider only a specific set of boundary conditions, but a whole class including, e.g., Dirichlet, Neumann, and Robin conditions. Our analysis will also cover interface conditions corresponding to a Finite Element Method - Boundary Element Method (FEM-BEM) coupling. In particular, we shall focus on three classical FEM-BEM couplings, namely the Costabel, Johnson-Nédélec and Bielak-MacCamy couplings. As a remarkable outcome, the present contribution yields well-posed substructured formulations of these classical FEM-BEM couplings for wavenumbers different from classical spurious resonances. We also establish an explicit relation between the dimensions of the kernels of the initial variational formulation, the local problems and the substructured formulation. That relation especially holds for any wavenumber for the substructured formulation of Costabel FEM-BEM coupling, which allows us to prove that the latter formulation is well-posed even at spurious resonances. Besides, we introduce a systematically geometrically convergent iterative method for the Costabel FEM-BEM coupling, with estimates on the convergence speed.
• Planning in Branch-and-Bound: Model-Based Reinforcement Learning for Exact Combinatorial Optimization
  
  • Strang Paul
  • Alès Zacharie
  • Bissuel Côme
  • Juan Olivier
  • Kedad-Sidhoum Safia
  • Rachelson Emmanuel
  , 2025. Mixed-Integer Linear Programming (MILP) lies at the core of many real-world combinatorial optimization (CO) problems, traditionally solved by branch-and-bound (B&amp;B). A key driver influencing B&amp;B solvers efficiency is the variable selection heuristic that guides branching decisions. Looking to move beyond static, hand-crafted heuristics, recent work has explored adapting traditional reinforcement learning (RL) algorithms to the B&amp;B setting, aiming to learn branching strategies tailored to specific MILP distributions. In parallel, RL agents have achieved remarkable success in board games, a very specific type of combinatorial problems, by leveraging environment simulators to plan via Monte Carlo Tree Search (MCTS). Building on these developments, we introduce Plan-and-Branch-and-Bound (PlanB&amp;B), a model-based reinforcement learning (MBRL) agent that leverages a learned internal model of the B&amp;B dynamics to discover improved branching strategies. Computational experiments empirically validate our approach, with our MBRL branching agent outperforming previous state-of-the-art RL methods across four standard MILP benchmarks.
• Early-Reverberation Imaging Functions for Bounded Elastic Domains
  
  • Ducasse Eric
  • Rodriguez Samuel
  • Bonnet Marc
  Acta Acustica, EDP Sciences, 2026, 10, pp.2. 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. (10.1051/aacus/2025069)
  DOI : 10.1051/aacus/2025069
• Multiscale modeling for a class of high-contrast heterogeneous sign-changing problems
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Jin Xingguang
  • Ye Changqing
  Journal of Computational Mathematics -International Edition-, Global Science Press, 2026. The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEMGMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the T-coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.
• Differentiability and Regularization of Parametric Convex Value Functions in Stochastic Multistage Optimization
  
  • Le Franc Adrien
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  Journal of Optimization Theory and Applications, Springer Verlag, 2026, 208 (3), pp.111. 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. (10.1007/s10957-025-02876-1)
  DOI : 10.1007/s10957-025-02876-1
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• 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, 2026, 20, pp.305-329. 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. (10.3934/ipi.2025028)
  DOI : 10.3934/ipi.2025028
• Exploring low-rank structure for an inverse scattering problem with far-field data
  
  • Zhou Yuyuan
  • Audibert Lorenzo
  • Meng Shixu
  • Zhang Bo
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2026, 86 (1), pp.160-186. The inverse scattering problem exhibits an inherent low-rank structure due to its ill-posed nature; however developing low-rank structures for the inverse scattering problem remains challenging. In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to a H\"{o}lder-logarithmic type stability estimate for arbitrary unknown functions, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability. (10.1137/24M1663922)
  DOI : 10.1137/24M1663922
• On some coupled local and nonlocal diffusion models
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2026. We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted $H^1$-semi norm to model the local part, while two different weighted $H^s$-semi norms, with $s \in (0, 1)$, are used to model the nonlocal part. The corresponding strong formulations are derived. In doing so, one needs to develop some technical tools, such as suitable integration by parts formulas for operators with variable diffusivity, and one also needs to study the mapping properties of the Neumann operators that arise. In contrast to problems coupling purely local models, in which one requires transmission conditions on the interface between the subdomains, the presence of a nonlocal operator may give rise to nonlocal fluxes. These nonlocal fluxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specific problem, that we consider most relevant, and study regularity of solutions and finite element discretizations. We provide numerical experiments to illustrate the most salient features of the models. (10.1142/S0218202526500442)
  DOI : 10.1142/S0218202526500442
• Trapped modes in electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  Integral Equations and Operator Theory, Springer Verlag, 2026. We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.
• A general-purpose global regularization method for 3D volume integral operators
  
  • Anderson Thomas G.
  • Bonnet Marc
  • Faria Luiz M.
  • Pérez-Arancibia Carlos
  , 2026. Singular volume integral operators associated with constant-coefficient partial differential operators extend the applicability of potential theory to inhomogeneous problems, for example arising from nonlinearities or variable coefficients. Typically the PDE kernels in these operators give rise to singularities at all O(1/h^3 ) volume discretization/evaluation points in a mesh of characteristic size h, while the slowly-decaying nature of such kernels give rise to long-range interactions that require coupling to fast summation algorithms. The presented method uses Green's identities to regularize a wide variety of both scalar-valued and vector-valued volume integral operators by use of a certain regularizing volume density interpolant. The analysis shows how the regularizing effect of the interpolant is global in the sense that the interpolation quality increases in an exactly compensatory fashion as the distance to the Green's function singularity decreases. High-order convergence estimates with tabulated simplex quadratures are established, including with exact representation of curved domains.