Publications

2026

• 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
• Waves within a network of slowly time-modulated interfaces: time-dependent effective properties, reciprocity and high-order dispersion
  
  • Darche Michaël
  • Lombard Bruno
  • Touboul Marie
  • Assier Raphaël
  • Guenneau Sebastien
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2026. We consider wave propagation through a 1D periodic network of slowly time-modulated interfaces. Each interface is modelled by time-dependent spring-mass jump conditions, where mass and rigidity interface parameters are modulated in time. Low-frequency homogenisation yields a leading-order model described by an effective time-dependent wave equation, i.e. a wave equation with effective mass density and Young's modulus which are homogeneous in space but depend on time. This means that time-dependent bulk effective properties can be created by an array where only interfaces are modulated in time. The occurrence of k-gaps in case of a periodic modulation is also analysed. Second-order homogenisation is then performed and leads to an effective model which is reciprocal but encapsulates higher-order dispersive effects. These findings and the limitations of the models are illustrated through time-domain simulations.
• Slip optimization on arbitrary 3D microswimmers: a reduced-dimension and boundary-integral framework
  
  • Bonnet Marc
  • Das Kausik
  • Veerapaneni Shravan
  • Zhu Hai
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2026. This article presents a computational framework for determining the optimal slip velocity of a microswimmer with arbitrary three-dimensional geometry suspended in a viscous fluid. The objective is to minimize the hydrodynamic power dissipation required to maintain unit speed along the net swimming direction. By exploiting the linearity of the Stokes equations and the Lorentz reciprocal theorem, we derive an explicit linear operator that maps the tangential surface slip velocity to the resulting rigid-body translational and rotational velocities, effectively decoupling the hydrodynamic boundary value problem from the optimization loop. The a priori infinite-dimensional search space for the slip optimization is reduced to the finite dimension $r$ of rigid-body motions by finding an appropriate subspace of the operator's domain. This reduces the PDE-constrained optimization to a low-dimensional programming problem that can be solved at negligible computational cost once the system matrices are assembled. The optimization algorithm requires 2$r$ auxiliary flow problems that are solved numerically using a high-order boundary integral method. We validate the accuracy of the proposed method and present optimal slip profiles and swimming trajectories for a variety of microswimmer shapes. We investigate the effect of some common geometrical symmetries of the swimmer shape on the resulting optimal motion, and in particular present a modified version of the slip optimization algorithm for axisymmetric shapes, where tangential rigid-body velocities may occur.
• Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions
  
  • Bohne Nis-Erik
  • Ciarlet Patrick
  • Sauter Stefan
  Foundations of Computational Mathematics, Springer Verlag, 2026. In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $(d-1)$ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does not exist in higher space dimension and $k>1$.
• Verification theorem related to a zero sum stochastic differential game, based on a chain rule for non-smooth functions
  
  • Ciccarella Carlo
  • Russo Francesco
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2026, 64 (1), pp.409-431. In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium. Suppose the validity of the classical Isaacs' condition and the existence of a (what is termed) quasi-strong solution to the Bellman-Isaacs (BI) equations. If the diffusion coefficient of the state equation is non-degenerate, we are able to show the existence of a saddle point constituted by a couple of feedback controls that achieve the value of the game: moreover, the latter is equal to the (necessarily unique) solution of the BI equations. A suitable generalization is available when the diffusion is possibly degenerate. Similarly we have also improved a well-known verification theorem in stochastic control theory. The techniques of stochastic calculus via regularization we use, in particular specific chain rules, are borrowed from a companion paper of the authors. (10.1137/24M1696676)
  DOI : 10.1137/24M1696676
• Machine Learning for Scientific Computing and Numerical Analysis
  
  • Dolean Victorita
  • Montanelli Hadrien
  , 2026. This MSc course introduces and develops advanced methods at the intersection of machine learning and scientific computing, with a special emphasis on solving and analyzing forward and inverse problems governed by partial differential equations. Students will learn how to combine classical numerical methods with modern neural-network architectures to approximate functions, operators, and solution maps, while critically assessing stability, generalization, and interpretability.
• THE L 2 -NORM OF THE FORWARD STOCHASTIC INTEGRAL W.R.T. FRACTIONAL BROWNIAN MOTION H > 1 2
  
  • Ohashi Alberto
  • Russo Francesco
  Bulletin des Sciences Mathématiques, Elsevier, 2026, 213 (213). In this article, we present the exact expression of the L 2-norm of the forward stochastic integral driven by the multi-dimensional fractional Brownian motion with parameter 1 2 < H < 1. The class of integrands only requires rather weak integrability conditions compatible w.r.t. a random finite measure whose density is expressed as a second-order polynomial of the underlying driving Gaussian noise. A simple consequence of our results is the exact expression of the L 2-norm for the pathwise Young integral. (10.1016/j.bulsci.2026.103881)
  DOI : 10.1016/j.bulsci.2026.103881
• Sampled-Based Guided Quantum Walk: Non-variational quantum algorithm for combinatorial optimization
  
  • Nzongani Ugo
  • Laplace Mermoud Dylan
  • Di Molfetta Giuseppe
  • Simonetto Andrea
  Quantum, Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2026, 10, pp.2234. We introduce SamBa-GQW, a novel quantum algorithm for solving binary combinatorial optimization problems of arbitrary degree with no use of any classical optimizer. The algorithm is based on a continuous-time quantum walk on the solution space represented as a graph. The walker explores the solution space to find its way to vertices that minimize the cost function of the optimization problem. The key novelty of our algorithm is an offline classical sampling protocol that gives information about the spectrum of the problem Hamiltonian. Then, the extracted information is used to guide the walker to high quality solutions via a quantum walk with a time-dependent hopping rate. We investigate the performance of SamBa-GQW on several quadratic problems, namely MaxCut, maximum independent set, portfolio optimization, and higher-order polynomial problems such as LABS, MAX-$k$-SAT and a quartic reformulation of the travelling salesperson problem. We empirically demonstrate that SamBa-GQW finds high quality approximate solutions on problems up to a size of $n=20$ qubits by only sampling $n^2$ states among $2^n$ possible decisions. SamBa-GQW compares very well also to other guided quantum walks and QAOA. (10.22331/q-2026-10-13-2234)
  DOI : 10.22331/q-2026-10-13-2234
• Solving numerically the two-dimensional time harmonic Maxwell problem with sign-changing coefficients
  
  • Chaaban Farah
  • Ciarlet Patrick
  • Rihani Mahran
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. We are investigating the numerical solution to the 2D time-harmonic Maxwell equations in the presence of a classical medium and a metamaterial, that is with sign-changing coefficients. As soon as the problem has a (unique) solution, we are able to build a converging numerical approximation based on the finite element method, for which there is no constraint on the meshes related to the sign-changing behavior. To that aim, we use Lagrange finite elements to approximate the scalar potentials appearing in the Helmholtz decomposition of the vector-valued electromagnetic fields. Convergence in strong norm is proven for the fields. Numerical examples illustrate the theory.
• Multiscale methods for wave propagation in materials with sign-changing coefficients
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Jin Xingguang
  • Ye Changqing
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2026. From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical theory, particularly when the effective dielectric permittivity and/or magnetic permeability are negative. This situation can transform a coercive operator into a non-coercive one, potentially leading to ill-posedness. In this paper, we utilize the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), specifically designed for time-harmonic electromagnetic wave problems, where the construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. Based on the framework of T-coercivity theory and resolution conditions, we establish the inf-sup stability and provide an a priori error estimate for the proposed method. The numerical results demonstrate the effectiveness and robustness of our approach in handling such sophisticated coefficient profiles.
• Well-posed homogenized strain-gradient models for linear elastodynamics and elastostatics in arbitrary periodic media
  
  • Cornaggia Rémi
  • Bonnet Marc
  • Rosi Giuseppe
  • El Ouafa Saad
  • Auffray Nicolas
  , 2026. This work develops well-posed homogenized strain-gradient models for linear elastostatics and elastodynamics in periodic media, with a primary focus on elastic wave propagation. Using the classical two-scale asymptotic expansion method, we carry out second-order periodic homogenization for media in $\Rbb^d$ ($d = 2, 3$), with no restriction on the periodicity cell geometry or material distribution. Reciprocity identities applied to suitably chosen pairs of cell solutions provide alternative expressions for the effective stiffness and inertia tensors arising at the leading, first and second orders, substantially reducing the number of cell problems that must actually be solved. Since direct two-scale homogenization beyond leading order generically yields ill-posed effective operators, a Boussinesq-trick procedure is introduced, involving a tunable scalar weight, to recast the resulting fourth-order partial differential equation as a valid strain-gradient elasticity (SGE) model possessing the requisite symmetry, sign-definiteness and coercivity properties. These properties are then used, via the Hille-Yosida theorem, to establish the well-posedness of the corresponding transient initial-value and forced-response problems. Several practically relevant special cases are examined, including centrosymmetric cells, homogeneous mass density and homogeneous elasticity, each yielding simplified model structures. Numerical illustrations on three two-dimensional periodicity cells (square, hexagonal and a non-centrosymmetric chiral lattice) compare the resulting dispersion relations against reference Floquet-Bloch computations and assess transient wave propagation, demonstrating the model's capacity to capture anisotropic and dispersive effects beyond classical elasticity while preserving mathematical well-posedness.
• Spectral theory for periodic operators
  
  • Fliss Sonia
  , 2026, 67. These lecture notes present the main results on the spectral theory of periodic operators with or without perturbations. After recalling the Floquet Bloch transform and the main results of spectral theory, we prove that periodic operators have a band spectrum with possibly band gaps. We will focus in particular on the so-called SSH discrete operator and 1D differential operators for which explicit computations can be done. We will then show few techniques to create gaps by playing with the coefficients. Finally, we will study the effect of perturbations of the periodic coefficients on the spectrum of the underlying operator. In particular, when the perturbation is localized, some discrete spectrum can appear. These lecture was given during the Research School "Théorie Spectrale, Contrôle et Problèmes inverses" in the framework of the CIRM-SMF Program.
• A global-in-time domain decomposition approach for transient acoustic-elastic interaction
  
  • Bonnet Marc
  • Chaillat Stéphanie
  • Nassor Alice
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2026. This work develops a global-in-time iterative domain decomposition approach for transient fluid-structure interaction problems involving acoustic scattering by elastic obstacles. The proposed method, inspired by optimized Schwarz waveform relaxation algorithms, proceeds by iteratively exchanging Robin boundary conditions, enabling the non-intrusive coupling of distinct fluid and structure solvers, and allows arbitrary transient incident acoustic fields. We prove the convergence of the proposed coupling iterations in continuous form. A BEM-FEM coupling implementation of the method is then validated against a reference analytical solution, and its efficiency, accuracy and robustness demonstrated through numerical experiments on configurations representative of potential industrial applications.
• Squirmers with arbitrary shape and slip: modeling, simulation, and optimization
  
  • Das Kausik
  • Zhu Hai
  • Bonnet Marc
  • Veerapaneni Shravan
  , 2026. We consider arbitrary-shaped microswimmers of spherical topology and propose a framework for expressing their slip velocity in terms of tangential basis functions defined on the boundary of the swimmer using the Helmholtz decomposition. Given a time-independent slip velocity profile, we show that the trajectory followed by the microswimmer is a circular helix. We derive analytical expressions for the translational and rotational velocities of a prolate spheroid swimmer in terms of its Helmholtz decomposition modes and explore the effect of aspect ratio on these rigid body velocities. Then, for a given arbitrary swimmer shape of spherical topology, we investigate which slip profile minimizes the total power loss. A partial minimization is performed in which the direction of net motion of the swimmer is prescribed, followed by a global optimization procedure in which the best net motion direction is determined. The optimization results suggest that the competition between linear and rotational optimal motion is linked to symmetries in the shape of the microswimmer.
• An entropy penalized approach for stochastic control problems. Complete version
  
  • Bourdais Thibaut
  • Oudjane Nadia
  • Russo Francesco
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2026, 64 (1), pp.363-386. In this paper, we propose an original approach to stochastic control problems. We consider a weak formulation that is written as an optimization (minimization) problem on the space of probability measures. We then introduce a penalized version of this problem obtained by splitting the minimization variables and penalizing the discrepancy between the two variables via an entropy term. We show that the penalized problem provides a good approximation of the original problem when the weight of the entropy penalization term is large enough. Moreover, the penalized problem has the advantage of giving rise to two optimization subproblems that are easy to solve in each of the two optimization variables when the other is fixed. We take advantage of this property to propose an alternating optimization procedure that converges to the infimum of the penalized problem with a rate $O(1/k)$, where $k$ is the number of iterations. The relevance of this approach is illustrated by solving a high-dimensional stochastic control problem aimed at controlling consumption in electrical systems. (10.1137/25M1741364)
  DOI : 10.1137/25M1741364
• Analysis of time-harmonic electromagnetic problems with elliptic material coefficients
  
  • Ciarlet Patrick
  • Modave Axel
  Mathematical Methods in the Applied Sciences, Wiley, 2026, 49, pp.3797-3815. 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
• A HDG method with transmission variables for time-harmonic wave propagation problems with constant coefficients
  
  • Pescuma Simone
  • Gabard Gwénaël
  • Chaumont-Frelet Théophile
  • Modave Axel
  , 2026. Iterative finite element solvers for time-harmonic wave problems are notoriously slow to converge, owing to fundamental properties of these problems. We present a variant of the hybridizable discontinuous Galerkin (HDG) method that is better suited to fast iterative solution. Unlike the standard hybridization strategy, which eliminates physical unknowns by introducing an auxiliary numerical flux on element faces, our approach instead introduces a transmission variable on those faces. For Helmholtz problems, this reformulation, known as CHDG, has been shown to significantly accelerate the convergence of iterative schemes relative to standard HDG. The present work extends CHDG to a general framework covering wave propagation problems with constant coefficients, capable of handling diverse wave types in a unified manner. We prove that the resulting hybridized system is well-posed and amenable to fixed-point iteration. As a practical application, we apply the method to the time-harmonic linearized Euler equations with a uniform subsonic mean flow. The method is illustrated through two-dimensional numerical benchmarks involving both sound and vorticity waves, with a systematic comparison of the convergence behaviour of several iterative schemes across a range of configurations.