Publications

2025

• Fast Boundary Element Methods Beyond Homogeneous Media: Challenges Towards Realistic Wave Propagation Simulations
  
  • Chaillat Stéphanie
  , 2026. <div><p>Boundary Element Methods (BEMs), based on the discretization of boundary integral equations, have proven particularly well-suited for modeling wave propagation in unbounded domains. In recent years, significant advances have made these methods applicable to realistic configurations. Fast algorithms, including the Fast Multipole Method, low-rank approximations, and mesh adaptivity, have significantly mitigated the inherent limitations of BEMs, substantially reducing computational costs and making BEM a competitive tool for large-scale simulations. While fast BEMs are now mature for homogeneous media and simple geometries, their applicability to more complex scenarios requires additional methodological advances. This review highlights recent collaborative efforts that expand BEMs to more challenging problems, including novel FEM-BEM coupling strategies inspired by volumetric domain decomposition methods, preconditioning techniques for multiple-scattering problems, and approaches for piecewise homogeneous media that extend BEM applicability beyond fully homogeneous domains.</p></div>
• Radial perfectly matched layers and infinite elements for the anisotropic wave equation
  
  • Halla Martin
  • Kachanovska Maryna
  • Wess Markus
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2025, 57 (3), pp.3171-3216. We consider the scalar anisotropic wave equation. Recently a convergence analysis for radial perfectly matched layers (PML) in the frequency domain was reported and in the present article we continue this approach into the time domain. First we explain why there is a good hope that radial complex scalings can overcome the instabilities of PML methods caused by anisotropic materials. Next we discuss some sensitive details, which seem like a paradox at the first glance: if the absorbing layer and the inhomogeneities are sufficiently separated, then the solution is indeed stable. However, for more general data the problem becomes unstable. In numerical computations we observe instabilities regardless of the position of the inhomogeneities, although the instabilities arise only for fine enough discretizations. As a remedy we propose a complex frequency shifted scaling and discretizations by Hardy space infinite elements or truncation-free PMLs. We show numerical experiments which confirm the stability and convergence of these methods. (10.1137/24M1636551)
  DOI : 10.1137/24M1636551
• Optimized Schwarz Methods in Time for Discrete Transport Control
  
  • Bui Duc-Quang
  • Delourme Bérangère
  • Halpern Laurence
  • Kwok Felix
  , 2025. We investigate optimized Schwarz domain decomposition methods in time for the control of the 1D transport equation. In the case of an internal control over the whole domain, the optimization problem can be transformed into a system of two coupled PDEs. We then apply the time-domain decomposition (without overlap) strategy on this PDE system as well as on its discretized counterpart. Under Fourier analysis, we analyse three different iterations: the fixed point iteration, the relaxed iteration and the preconditioned GMRES method. For each case, we propose parameters for the transmission conditions that lead to fast convergence of the method. We illustrate our results by numerical examples.
• A multi-objective optimization approach for generalized linear multiplicative programming
  
  • Nguyen Minh Hieu
  • Nguyen Thanh Loan
  , 2026, 1688. Multiplicative programming is a fundamental mathematical optimization problem in which the objective function contains a product of several real-valued functions. This paper deals with a class of multiplicative programming, called generalized linear multiplicative programming (GLMP), in which the objective is to minimize the product of two positive linear functions with general positive powers under linear constraints. Since the objective is a typical non-convex function, GLMP may have multiple local minima, making it computationally challenging. To address this, we propose a multi-objective optimization-based approach. By treating each function as an objective to be minimized, we show that a solution of GLMP is necessarily a non-dominated extreme point located on the vertex of the convex hull of the Pareto front. Then, we use a recursive algorithm to determine the set of all non-dominated extreme points. Notice that the solutions of GLMP can be directly extracted from this set. Furthermore, based on the Weighted Sum Method, it requires only solving one linear program in each iteration. Finally, we provide computational results on a specific instance of GLMP with 0-1 knapsack constraints, indicating that our approach is promising. (10.1007/978-3-032-08381-4_8)
  DOI : 10.1007/978-3-032-08381-4_8
• Homogenization of stable-like operators with random, ergodic coefficients
  
  • Klimsiak Tomasz
  • Komorowski Tomasz
  • Marino Lorenzo
  Journal of Differential Equations, Elsevier, 2025, 430, pp.113183. We show homogenization for a family of R d -valued stable-like processes (X ε;θ t ) t≥0 , ε ∈ (0, 1], whose (random) Fourier symbols equal q ε (x, ξ; θ) = 1 ε α q x ε , εξ; θ , where<p>1 -e iy•ξ + iy • ξ1 {|y|≤1} a(x; θ)y, y |y| d+2+α dy, for (x, ξ, θ) ∈ R 2d × Θ. Here α ∈ (0, 2) and the family (a(x; θ)) x∈R d of d × d symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space (Θ, H, m). We assume that the random field is deterministically bounded and non-degenerate, i.e. |a(x; θ)| ≤ Λ and Tr(a(x; θ)) ≥ λ for some Λ, λ &gt; 0 and all θ ∈ Θ. In addition, we suppose that the field is regular enough so that for any θ ∈ Θ, the operator -q(•, D; θ), defined on the space of compactly supported C 2 functions on R d , is closable in the space of continuous functions vanishing at infinity and its closure generates a Feller semigroup. We prove the weak convergence of the laws of (X ε;θ t ) t≥0 , as ε ↓ 0, in the Skorokhod space, m-a.s. in θ, to an α-stable process whose Fourier symbol q(ξ) is given by q(ξ) = Ω q(0, ξ; θ)Φ * (θ) m(dθ), where Φ * is a strictly positive density w.r.t. measure m. Our result has an analytic interpretation in terms of the convergence, as ε ↓ 0, of the solutions to random integro-differential equations ∂ t u ε (t, x; θ) = -q ε (x, D; θ)u ε (t, x; θ), with the initial condition u ε (0, x; θ) = f (x), where f is a bounded and continuous function on R d .</p> (10.1016/j.jde.2025.02.054)
  DOI : 10.1016/j.jde.2025.02.054
• Convergence analysis of GMRES applied to Helmholtz problems near resonances
  
  • Dolean Victorita
  • Marchand Pierre
  • Modave Axel
  • Raynaud Timothée
  , 2025. In this work we study how the convergence rate of GMRES is influenced by the properties of linear systems arising from Helmholtz problems near resonances or quasi-resonances. We extend an existing convergence bound to demonstrate that the approximation of small eigenvalues by harmonic Ritz values plays a key role in convergence behavior. Next, we analyze the impact of deflation using carefully selected vectors and combine this with a Complex Shifted Laplacian preconditioner. Finally, we apply these tools to two numerical examples near (quasi-)resonant frequencies, using them to explain how the convergence rate evolves.
• On some coupled local and nonlocal diffusion models
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  , 2025. 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.
• Open Review of "Normal form analysis of nonlinear oscillator equations with automated arbitrary order expansions
  
  • de Figueiredo Stabile André
  • Touzé Cyril
  • Vizzaccaro Alessandra
  • Römer Ulrich
  • Raze Ghislain
  • Chaillat Stéphanie
  , 2025.
• Time-harmonic wave propagation in junctions of two periodic half-spaces
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2025, 7 (2), pp.299-357. We are interested in the Helmholtz equation in a junction of two periodic half-spaces. When the overall medium is periodic in the direction of the interface, Fliss and Joly (2019) proposed a method which consists in applying a partial Floquet-Bloch transform along the interface, to obtain a family of waveguide problems parameterized by the Floquet variable. In this paper, we consider two model configurations where the medium is no longer periodic in the direction of the interface. Inspired by the works of Gérard-Varet and Masmoudi (2011, 2012), and Blanc, Le Bris, and Lions (2015), we use the fact that the overall medium has a so-called quasiperiodic structure, in the sense that it is the restriction of a higher dimensional periodic medium. Accordingly, the Helmholtz equation is lifted onto a higher dimensional problem with coefficients that are periodic along the interface. This periodicity property allows us to adapt the tools previously developed for periodic media. However, the augmented PDE is elliptically degenerate (in the sense of the principal part of its differential operator) and thus more delicate to analyse. (10.2140/paa.2025.7.299)
  DOI : 10.2140/paa.2025.7.299
• Méthode hybride de simulation de champs ultrasonores dans une grande structure stratifiée avec des objets au contact
  
  • Kubecki Romain
  • Ducasse Eric
  • Bonnet Marc
  • Deschamps Marc
  , 2025. Ce travail a pour objectif de simuler la propagation d'ultrasons dans une structure stratifiée de grande taille comportant des objets au contact (de type traducteur, raidisseur, ou autre), dans un contexte de contrôle non destructif. La taille modérée des objets permet leur simulation par éléments finis, qui est par contre prohibitive pour la structure stratifiée de base. Si cette dernière est de géométrie canonique (plane ou tubulaire à symétrie de révolution), les champs peuvent en revanche être calculés par une méthode semi-analytique rapide utilisant des transformées de Laplace en temps et de Fourier par rapport aux coordonnées « longitudinales » (plan de la plaque ou positions axiale et azimutale dans le tube). En effet, dans le domaine <latex>(k,r,s)</latex> (<latex>k~vecteur</latex> d'onde, <latex>r~position</latex> dans l'épaisseur, <latex>s~variable</latex> de Laplace), le problème de propagation peut être résolu de manière exacte, et massivement parallélisable. Pour exploiter les atouts des deux méthodes, nous proposons une approche itérative de couplage par <i>décomposition de domaine</i> (DDM), reposant sur une suite de problèmes de propagation dans chaque sous-domaine comprenant sur leur interface commune des conditions aux limites dépendant des solutions de l'itération précédente. La littérature montre que le choix de conditions de Robin (de type impédance) entre deux domaines couplés garantit dans beaucoup de situations la convergence des itérations de couplage. Nous prouvons que cette convergence a bien lieu pour notre contexte particulier et présentons une validation numérique préliminaire en configuration 2D. Le caractère spatialement non-local du traitement semi-analytique de la structure stratifiée nous conduit ensuite à construire des fonctions de base négligeables en-dehors d'un voisinage de l'interface et à développer un protocole spécifique pour leur couplage avec les éléments finis. Ces deux aspects constituent les principaux ingrédients de la méthode hybride proposée ici. <latex>\medskip\hspace20mm</latex><i>Ce travail est financé par la DGA-AID et le CEA-List.</i>
• Optimisation par méthode adjointe discrète du bruit tonal d'une hélice estimé par la formulation fréquentielle de Hanson et Parzych
  
  • Mohammedi Yacine
  • Daroukh Majd
  • Buszyk Martin
  • Hajczak Antoine
  • Salah El-Din Itham
  • Bonnet Marc
  , 2025. Ce travail est consacré à l'optimisation à visée aéroacoustique de la forme d'une pale d’hélice en utilisant la méthode adjointe discrète. Cette dernière sera appliquée aux équations de Navier-Stokes stationnaires ainsi qu'à la formulation intégrale fréquentielle de Hanson et Parzych destinée au calcul du bruit tonal de rotor. Les sensibilités de la pression acoustique sont obtenues par dérivation analytique de la formulation intégrale. Ainsi, les sensibilités de toute fonction objectif exprimée en fonction de la pression acoustique peuvent être calculées. Ensuite, un solveur adjoint discret des équations de Navier-Stokes avec moyenne de Reynolds fournit les gradients de la fonction objectif en fonction des paramètres de forme. Ces derniers sont validés par comparaison avec une estimation par différences finies précise à l'ordre deux. Enfin, une optimisation multidisciplinaire et multi-objectifs est effectuée sur une hélice tripale subsonique isolée en condition de vol de croisière.
• Roadmap on metamaterial theory, modelling and design
  
  • Davies Bryn
  • Szyniszewski Stefan
  • Dias Marcelo
  • de Waal Leo
  • Kisil Anastasia
  • P Smyshlyaev Valery
  • Cooper Shane
  • Kamotski Ilia
  • Touboul Marie
  • Craster Richard
  • Capers James
  • Horsley Simon
  • Hewson Robert
  • Santer Matthew
  • Murphy Ryan
  • Thillaithevan Dilaksan
  • Berry Simon
  • Conduit Gareth
  • Earnshaw Jacob
  • Syrotiuk Nicholas
  • Duncan Oliver
  • Kaczmarczyk Łukasz
  • Scarpa Fabrizio
  • Pendry John
  • Martí-Sabaté Marc
  • Guenneau Sébastien
  • Torrent Daniel
  • Cherkaev Elena
  • Wellander Niklas
  • Alù Andrea
  • Madine Katie
  • Colquitt Daniel
  • Sheng Ping
  • Bennetts Luke
  • Krushynska Anastasiia
  • Zhang Zhaohang
  • Mirzaali Mohammad
  • Zadpoor Amir
  Journal of Physics D: Applied Physics, IOP Publishing, 2025, 58 (20), pp.203002. This Roadmap surveys the diversity of different approaches for characterising, modelling and designing metamaterials. It contains articles covering the wide range of physical settings in which metamaterials have been realised, from acoustics and electromagnetics to water waves and mechanical systems. In doing so, we highlight synergies between the many different physical domains and identify commonality between the main challenges. The articles also survey a variety of different strategies and philosophies, from analytic methods such as classical homogenisation to numerical optimisation and data-driven approaches. We highlight how the challenging and many-degree-of-freedom nature of metamaterial design problems call for techniques to be used in partnership, such that physical modelling and intuition can be combined with the computational might of modern optimisation and machine learning to facilitate future breakthroughs in the field. (10.1088/1361-6463/adc271)
  DOI : 10.1088/1361-6463/adc271
• Beyond the Fermat optimality rules
  
  • Abbasi Malek
  • Grad Sorin-Mihai
  • Théra Michel
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2025, 35 (2), pp.818-841. This work proposes a general framework for analyzing the behavior at its extrema of an extended real-valued function assumed neither convex nor differentiable and for which the classical Fermat rules of optimality do not apply. The tools used for building this frame are the notions of sup-subdifferential, recently introduced by two of the authors together with A. Kruger, and partial sup-subdifferentials. The sup-subdifferential is always a \textit{nonempty} enlargement of the Moreau-Rockafellar subdifferential from convex optimization. It satisfies most of the fundamental properties of the Moreau-Rockafellar subdifferential and enjoys certain calculus rules. The partial sup-subdifferentials are obtained by breaking down the sup-subdifferential into one-dimensional components through basis elements and play the same role as the partial derivatives in the Fermat optimality rules. (10.1137/23M1578036)
  DOI : 10.1137/23M1578036
• An inverse problem related to an elasto-plastic beam
  
  • Bourgeois Laurent
  • Mercier Jean-François
  Inverse Problems, IOP Publishing, 2025, 41 (10). We consider an elasto-plastic beam and address the following inverse problem: external forces have created some plastic strains in this beam, which therefore shows a residual observable deformation once the structure is load-free. Can we retrieve the loading history from this observation, or at least the plastic strains ? After proving the well-posedness of the forward problem, we show that the solution can be described in a semi-explicit way in the pure bending case, so that the forward problem amounts to a one dimensional non linear problem. Such problem is smooth enough for us to solve the inverse problem by using a classical least square method, which is illustrated with the help of some numerical examples. (10.1088/1361-6420/ae0e49)
  DOI : 10.1088/1361-6420/ae0e49
• An entropy penalized approach for stochastic optimization with marginal law constraints. Complete version
  
  • Bourdais Thibaut
  • Oudjane Nadia
  • Russo Francesco
  , 2025. This paper focuses on stochastic optimal control problems with constraints in law, which are rewritten as optimization (minimization) of probability measures problem on the canonical space. We introduce a penalized version of this type of problems by splitting the optimization variable and adding an entropic penalization term. We prove that this penalized version constitutes a good approximation of the original control problem and we provide an alternating procedure which converges, under a so called "Stability Condition", to an approximate solution of the original problem. We extend the approach introduced in a previous paper of the same authors including a jump dynamics, non-convex costs and constraints on the marginal laws of the controlled process. The interest of our approach is illustrated by numerical simulations related to demand-side management problems arising in power systems.
• A Production Routing Problem with Mobile Inventories
  
  • Lefgoum Raian
  • Afsar Sezin
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2025. <div><p>Hydrogen is an energy vector, and one possible way to reduce CO 2 emissions. This paper focuses on a hydrogen transport problem where mobile storage units are moved by trucks between sources to be refilled and destinations to meet demands, involving swap operations upon arrival. This contrasts with existing literature where inventories remain stationary. The objective is to optimize daily routing and refilling schedules of the mobile storages. We model the problem as a flow problem on a time-expanded graph, where each node of the graph is indexed by a time-interval and a location and then, we give an equivalent Mixed Integer Linear Programming (MILP) formulation of the problem. For small to medium-sized instances, this formulation can be efficiently solved using standard MILP solvers. However, for larger instances, the computational complexity increases significantly due to the highly combinatorial nature of the refilling process at the sources. To address this challenge, we propose a two-step heuristic that enhances</p></div>
• Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy trade-offs
  
  • Zylberman Julien
  • Nzongani Ugo
  • Simonetto Andrea
  • Debbasch Fabrice
  ACM Transactions on Quantum Computing, ACM, 2025. Unitary and non-unitary diagonal operators are fundamental building blocks in quantum algorithms with applications in the resolution of partial differential equations, Hamiltonian simulations, the loading of classical data on quantum computers (quantum state preparation) and many others. In this paper, we introduce a general approach to implement unitary and non-unitary diagonal operators with efficient-adjustable-depth quantum circuits. The depth, i.e. the number of layers of quantum gates of the quantum circuit, is reducible with respect either to the width, i.e. the number of ancilla qubits, or to the accuracy between the implemented operator and the target one. While exact methods have an optimal exponential scaling either in terms of size, i.e. the total number of primitive quantum gates, or width, approximate methods prove to be efficient for the class of diagonal operators depending on smooth, at least differentiable, functions. Our approach is general enough to allow any method for diagonal operators to become adjustable-depth or approximate, decreasing the depth of the circuit by increasing its width or its approximation level. This feature offers flexibility and can match with the hardware limitations in coherence time or cumulative gate error. We illustrate these methods by performing quantum state preparation and non-unitary-real-space simulation of the diffusion equation: an initial Gaussian function is prepared on a set of qubits before being evolved through the non-unitary evolution operator of the diffusion process. (10.1145/3718348)
  DOI : 10.1145/3718348
• Mathematical and numerical analysis of the modes of a heterogeneous electromagnetic waveguide.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2025. In the homogeneous case, i.e. with constant epsilon and mu, the modes (E_n, H_n, \beta_n) are easily obtained by solving scalar problems in the section S of the guide and are pairwise orthogonal in L^2(S). They are either propagating (\beta in R) or purely evanescent (\beta in iR) and they have phase and group velocities of the same sign. For heterogeneous guides, i.e. with varying epsilon and mu in the section, these properties are generally not true and the mathematical analysis of the modes is much more delicate. In this talk, we present different formulations to study them and discuss their respective advantages. For strong variations of epsilon and/or mu, we show numerically that inverse modes, with group and phase velocities of opposite sign, can exist. Such cases for which PMLs fail to capture the outgoing solution are one of the reasons why we develop modal transparent conditions.
• Not all sub-Riemannian minimizing geodesics are smooth
  
  • Chitour Yacine
  • Jean Frédéric
  • Monti Roberto
  • Rifford Ludovic
  • Sacchelli Ludovic
  • Sigalotti Mario
  • Socionovo Alessandro
  , 2025. A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a C 2 but not C 3 length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.
• Integral equation methods for acoustic scattering by fractals
  
  • Caetano António
  • Chandler-Wilde Simon
  • Claeys Xavier
  • Gibbs Andrew
  • Hewett David
  • Moiola Andrea
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2025, 481 (2306). We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer Γ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on Γ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when Γ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When Γ is uniformly of d -dimensional Hausdorff dimension in a sense we make precise (a d -set), the operator in our equation is an integral operator on Γ with respect to d -dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When Γ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on Γ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code. (10.1098/rspa.2023.0650)
  DOI : 10.1098/rspa.2023.0650
• Alpha Mesh Swc: automatic and robust surface mesh generation from the skeleton description of brain cells
  
  • Mcsweeney-Davis Alex
  • Fang Chengran
  • Caruyer Emmanuel
  • Kerbrat Anne
  • Li Jing-Rebecca
  Briefings in Bioinformatics, Oxford University Press (OUP), 2025, 26 (3). In recent years, there has been a significant increase in publicly available skeleton descriptions of real brain cells from laboratories all over the world. In theory, this should make it possible to perform large scale realistic simulations on brain cells. However, currently there is still a gap between the skeleton descriptions and high quality simulation-ready surface and volume meshes of brain cells. We propose and implement a tool called {\it Alpha\_Mesh\_Swc} to generate automatically and efficiently triangular surface meshes that are optimized for finite elements simulations. We use an Alpha Wrapping method with an offset parameter on component surface meshes to efficiently generate a global watertight mesh. Then mesh simplification and re-meshing are used to produce an optimal surface mesh. Our methodology limits the number of surface triangles while preserving geometrical accuracy, permits cutting and gluing of cell components, is robust to imperfect skeleton descriptions, and allows mixed cell descriptions (surface meshes combined with skeletons). We compared the robustness, performance and accuracy of {\it Alpha\_Mesh\_Swc} against existing tools and found significant improvement in terms of mesh accuracy. We show, on average, we can generate fully automatically a brain cell (neurons or glia) surface mesh in a couple of minutes on a laptop computer resulting in a simplified surface mesh with only around 10k nodes. The resulting meshes were used to perform diffusion MRI simulations in neurons and microglia. The code and a number of sample brain cell surface meshes have been made publicly available. (10.1093/bib/bbaf258)
  DOI : 10.1093/bib/bbaf258
• A Two-Timescale Decision-Hazard-Decision Formulation for Storage Usage Values Calculation
  
  • Martinez Parra Camila
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Janin Jean-Marc
  • Ruiz Manuel
  , 2024. The penetration of renewable energies requires additional storages to deal with intermittency. Accordingly, there is growing interest in evaluating the opportunity cost (usage value) associated with stored energy in large storages, a cost obtained by solving a multistage stochastic optimization problem. Today, to compute usage values under uncertainties, an adequacy resource problem is solved using stochastic dynamic programming assuming a hazard-decision information structure. This modelling assumes complete knowledge of the coming week uncertainties, which is not adapted to the system operation as the intermittency occurs at smaller timescale. We equip the twotimescale problem with a new information structure considering planning and recourse decisions: decision-hazard-decision. This structure is used to decompose the multistage decision-making process into a nonanticipative planning step in which the on/off decisions for the thermal units are made, and a recourse step in which the power modulation decisions are made once the uncertainties have been disclosed. In a numerical case, we illustrate how usage values are sensitive as how the disclosure of information is modelled. (10.48550/arXiv.2408.17113)
  DOI : 10.48550/arXiv.2408.17113
• Differentiable Optimisation: Theory and Algorithms -- Part II: Algorithms
  
  • Simonetto Andrea
  , 2025. This course follows naturally OPT201, which covers the theory part of continuous optimisation. OPT201 focuses on optimality conditions, convexity, and duality. In OPT202, we will look at how to use these notions to build algorithms that solve the problems. In particular, the aim of the course is to be able to answer the questions, 1. Given an optimisation problem, which algorithm do I use to solve it? 2. Which properties and theoretical guarantees does the algorithm that I have chosen have? 3. Conversely, if I want to use a certain algorithm, which characteristics does the optimisation problem need to have? In order to answer to these three questions, we will need to build a theory of algorithms, and ultimately understand what we really mean by solving an optimisation problem.
• Energy stable and linearly well-balanced numerical schemes for the non-linear Shallow Water equations with Coriolis force
  
  • Audusse Emmanuel
  • Dubos Virgile
  • Gaveau Noémie
  • Penel Yohan
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2025, 47 (01), pp.A1-A23. We analyse a class of energy-stable and linearly well-balanced numerical schemes dedicated to the nonlinear Shallow Water equations with Coriolis force. The proposed algorithms rely on colocated finite-difference approx- imations formulated on cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semi-discrete energy estimates. Among the proposed algorithms a colocated finite-volume scheme is described. Numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes. (10.1137/22M1515707)
  DOI : 10.1137/22M1515707
• Global solution of quadratic problems using 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