Publications

2026

• 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&B). A key driver influencing B&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&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&B), a model-based reinforcement learning (MBRL) agent that leverages a learned internal model of the B&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
• 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.
• Stochastic transport by Gaussian noise
  
  • Flandoli Franco
  • Russo Francesco
  Rendiconti Lincei. Matematica e Applicazioni, European Mathematical Society, 2026. Diffusion with stochastic transport is investigated here when the random driving process is a very general Gaussian process, including Fractional Brownian motion. The purpose is the comparison with a deterministic PDE, which in certain cases represents the equation for the mean value. From this equation we observe a reduced dissipation property for small times and an enhanced diffusion for large times, with respect to delta correlated noise when regularity is higher than the one of Brownian motion, a fact interpreted qualitatively here as a signature of the modified dissipation observed for 2D turbulent fluids due to the inverse cascade. We give results also for the variance of the solution and for a scaling limit of a two-component noise input. (10.4171/RLM/1090)
  DOI : 10.4171/RLM/1090
• Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces
  
  • Issoglio Elena
  • Pagliarani Stefano
  • Russo Francesco
  • Trevisani Davide
  Journal of Differential Equations, Elsevier, 2026, 477, pp.114567. The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak Hörmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates. (10.1016/j.jde.2026.114567)
  DOI : 10.1016/j.jde.2026.114567
• Predicting topologically protected interface state with high-frequency homogenization
  
  • Touboul Marie
  • Lombard Bruno
  • Coutant Antonin
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2026, 354, pp.269-291. When two semi-infinite periodic media are joined together, a localized interface mode may exist, whose frequency belongs to their common band gap. Moreover, if certain spatial symmetries are satisfied, this mode is topologically protected and thus is robust to defects. A method has recently been proposed to identify the existence and the frequency of this mode, based on the computation of surface impedances at all the frequencies in the gap. In this work, we approximate the surface impedances thanks to highfrequency effective models, and therefore get a prediction of topologically protected interface states while only computing the solution of an eigenvalue problem at the edges of the bandgaps. We also show that the nearby eigenvalues high-frequency effective models give rise to a better approximation of the surface impedance.
• Concentration inequalities for semidefinite least squares based on data
  
  • Fabiani Filippo
  • Simonetto Andrea
  IEEE Signal Processing Letters, Institute of Electrical and Electronics Engineers, 2026, 33, pp.326-330. We study data-driven least squares (LS) problems with semidefinite (SD) constraints and derive finite-sample guarantees on the spectrum of their optimal solutions when these constraints are relaxed. In particular, we provide a high confidence bound allowing one to solve a simpler program in place of the full SDLS problem, while ensuring that the eigenvalues of the resulting solution are $\varepsilon$-close of those enforced by the SD constraints. The developed certificate, which consistently shrinks as the number of data increases, turns out to be easy-to-compute, distribution-free, and only requires independent and identically distributed samples. Moreover, when the SDLS is used to learn an unknown quadratic function, we establish bounds on the error between a gradient descent iterate minimizing the surrogate cost obtained with no SD constraints and the true minimizer. (10.1109/LSP.2025.3643385)
  DOI : 10.1109/LSP.2025.3643385
• 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.
• 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.
• 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
• 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
• 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
• Aeroacoustic Optimization of Transonic Open Rotors Using a Discrete Adjoint Framework
  
  • Mohammedi Yacine
  • Daroukh Majd
  • Buszyk Martin
  • Hajczak Antoine
  • Bonnet Marc
  • Salah El-Din Itham
  AIAA Journal, American Institute of Aeronautics and Astronautics, 2026. A discrete adjoint framework is developed to optimize rotor aerodynamic performance and tonal noise from steady-state simulations in the rotating frame of reference. Aerodynamic performance is characterized through the thrust and the torque, while the acoustic metric is determined using a simplified expression of the off-blade frequency-domain Ffowcs-Williams and Hawkings (FW-H) equation for far-field observers, following the model of Hanson and Parzych (1993) originally formulated for on-blade surfaces. This formulation is implemented and validated against an established time-domain FW-H solver.~Analytical expressions are derived for the sensitivities of aerodynamic quantities and far-field acoustic pressure, which enables the reconstruction of any objective function expressed in terms of acoustic pressure, such as acoustic power. These analytical sensitivities are successfully validated by comparison to second-order accurate finite-difference (FD) approximations. A mesh deformation tool updates the geometry and computes the corresponding gradient of the volume mesh. The discrete adjoint of a Reynolds-Averaged Navier–Stokes solver is then used to provide the sensitivities of the objective functions with respect to the blade shape parameters. These sensitivities are supplied to the optimizer, which updates the design parameters accordingly, and the mesh deformation tool subsequently modifies the geometry of the blade based on the optimizer’s feedback, thereby completing the gradient-based aeroacoustic design workflow. The optimization targets a composite objective function that combines torque and acoustic power at the Blade Passing Frequency (BPF), while enforcing thrust as a constraint. The whole process is applied on an academic isolated open rotor configuration at transonic conditions with equal weighting assigned to aerodynamic and acoustic objectives, resulting in a combined reduction of approximately 7% in torque and 13dB in acoustic power. Unsteady assessments of both the isolated rotor and rotor-stator configurations further show that the optimization reduces the dominant rotor noise without increasing the second and third BPF levels, while the remaining unsteady rotor and rotor-stator interaction sources represent opportunities for further noise reduction.
• 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>
• Boundary integral equation and convolution quadrature methods for transient problems: an introduction
  
  • Bonnet Marc
  • Chaillat Stéphanie
  , 2026. Boundary integral equation (BIE) methods are well-suited for the numerical solution of a wide range of linear PDEs, including evolution problems arising in acoustics, elastodynamics, electromagnetism, heat transfer, and related fields. While time-domain BIE formulations have a long history, the development of efficient numerical methods for their solution is more recent. Classical time-stepping BIE methods based on retarded potentials raise well-documented issues, which prompted the more-recent emergence of alternative approaches based on the concept of convolution quadrature (CQ). This review paper aims at providing an accessible and tutorialstyle presentation of CQ-based methods for time-domain BIEs. These exist in two main forms, namely the sequential-in-time CQ-BEM and a recent simultaneous-in-time form herein referred to as the Z-BEM. Both forms are based on adopting at the outset a time-discrete setting. Consequently, as informally shown in this paper, both approaches may be derived and described in an unified manner by means of the Z-transform, a discrete-time analog of the Laplace transform, with the linear evolution problem to be solved initially set either in PDE form or as a continuous time-domain BIE. Computational aspects, including algorithmic parameters and high-frequency approximations, are also discussed, together with comparative remarks on the Z-BEM and CQ-BEM approaches and a concise survey of the main developments and applications.
• 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
• 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.
• 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 $k&gt;1$.
• 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 &gt; 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 &lt; H &lt; 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