Publications

2021

• Stochastic incremental mirror descent algorithms with Nesterov smoothing
  
  • Grad Sorin-Mihai
  • Bitterlich Sandy
  , 2021. We propose a stochastic incremental mirror descent method constructed by means of the Nesterov smoothing for minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in a Euclidean space. The algorithm can be adapted in order to minimize (in the same setting) a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Another modification of the scheme leads to a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing for minimizing the sum of finitely many proper, convex and lower semicontinuous functions with a prox-friendly proper, convex and lower semicontinuous function in the same framework. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements.
• Stochastic incremental mirror descent algorithms with Nesterov smoothing
  
  • Grad Sorin-Mihai
  • Bitterlich Sandy
  , 2021. We propose a stochastic incremental mirror descent method constructed by means of the Nesterov smoothing for minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in a Euclidean space. The algorithm can be adapted in order to minimize (in the same setting) a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Another modification of the scheme leads to a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing for minimizing the sum of finitely many proper, convex and lower semicontinuous functions with a prox-friendly proper, convex and lower semicontinuous function in the same framework. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements.
• Analysis of time-harmonic electromagnetic problems in elliptic anisotropic media
  
  • Chicaud Damien
  , 2021. The numerical simulation of electromagnetic problems in complex physical settings is a trending topic which conveys many scientific and industrial applications, such as the design of optical metamaterials, or the study of cold plasmas. The mathematical and numerical analysis of Maxwell problems is wellknown in simple physical contexts, when the material parameters are isotropic. Some results in anisotropic media exist, but they generally tend to focus on the case where the material tensors are real symmetric (or complex) Hermitian) definite positive. However, problems in more complex media are not covered by the standard theory. Therefore, new mathematical tools need to be developped to analyse thses problems. This thesis aims at analysing time-harmonic electromagnetic problems for a general class of complex anisotropic material tensors. These are called ellopptic materials. We derive an extended functional framework well-suited for these anisotropic problems, generalizing well-known results. We study the well-posedness of Maxwell boundary value problems for Dirichlet, Neumann, and Robin boundary conditions. For the Robin case, the characterization of appropriate function spaces for Robin traces is addressed. The regularity of the solution and its curl is studied, and elements of numerical analysis for edge finite elements are provided. In the perspective of the use of Domain Decomposition Methods (DDM) for accelerated numerical computing, various decomposed formulations are proposed and studied, focusing on their right meaning in terms of function spaces and equivalence with the global problem. These results are complemented with some numerical DDM experimentations in anisotropic media.
• Smoothness of densities for path-dependent SDEs under Hörmander's condition
  
  • Ohashi Alberto
  • Russo Francesco
  • Shamarova Evelina
  Journal of Functional Analysis, Elsevier, 2021, 281 (11), pp.109225. We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given n-dimensional path-dependent SDE into a suitable Lp-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hörmander’s bracket condition in Rnfor non-anticipative SDE coefficients defining the Lie brackets in terms of vertical derivatives in the sense of the functional Itô calculus. Our pathway to the main result engages an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path techniques. (10.1016/j.jfa.2021.109225)
  DOI : 10.1016/j.jfa.2021.109225
• Smoothness of densities for path-dependent SDEs under Hörmander's condition
  
  • Ohashi Alberto
  • Russo Francesco
  • Shamarova Evelina
  Journal of Functional Analysis, Elsevier, 2021, 281 (11), pp.109225. We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given n-dimensional path-dependent SDE into a suitable Lp-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hörmander’s bracket condition in Rnfor non-anticipative SDE coefficients defining the Lie brackets in terms of vertical derivatives in the sense of the functional Itô calculus. Our pathway to the main result engages an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path techniques. (10.1016/j.jfa.2021.109225)
  DOI : 10.1016/j.jfa.2021.109225
• Mixed Integer Nonlinear Approaches for the Satellite Constellation Design Problem
  
  • Mencarelli Luca
  • Floquet Julien
  • Georges Frederic
  , 2021.
• Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure
  
  • Grad Sorin-Mihai
  • Boţ Radu Ioan
  • Meier Dennis
  • Staudigl Mathias
  , 2021. Zeros of the sum of a maximally monotone operator with a single-valued monotone one can be obtained as weak limits of trajectories of dynamical systems, for strong convergence demanding hypotheses being imposed. We extend an approach due to Attouch, Cominetti and coauthors, where zeros of maximally monotone operators are obtained as strong limits of trajectories of Tikhonov regularized dynamical systems, to forward-backward and forward-backward-forward dynamical systems whose trajectories strongly converge towards zeros of such sums of monotone operators under reasonable assumptions.
• New Methods of Isochrone Mechanics
  
  • Ramond Paul
  • Perez Jérôme
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2021, 62, pp.112704. Isochrone potentials, as defined by Michel H\'enon in the fifties, are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Isochrone potentials encompass the Kepler and harmonic potential, along with many other. In this article, we revisit the classical problem of motion in isochrone potentials, from the point of view of Hamiltonian mechanics. First, we use a particularly well-suited set of action-angle coordinates to solve the dynamics, showing that the well-known Kepler equation and eccentric anomaly parametrisation are valid for any isochrone orbit (and not just Keplerian ellipses). Second, by using the powerful machinery of Birkhoff normal forms, we provide a self-consistent proof of the isochrone theorem, that relates isochrone potentials to parabolae in the plane, which is the basis of all literature on the subject. Along the way, we show how some fundamental results of celestial mechanics such as the Bertrand theorem and Kepler's third law are naturally encoded in the formalism. (10.1063/5.0056957)
  DOI : 10.1063/5.0056957
• Outils élémentaires d'analyse pour les Equations aux Dérivées Partielles
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Bourgeois Laurent
  • Hazard Christophe
  , 2021, pp.132. Ce cours présente quelques-uns des principaux outils de l’analyse pour l’étude mathématique des équations aux dérivées partielles issues des sciences physiques ou humaines. Le chapitre 1 rappelle quelques notions essentielles concernant la topologie des espaces vectoriels normés. Le chapitre 2 est une introduction succincte à l’intégrale de Lebesgue, l’idée étant ici de comprendre quels sont les avantages de cette intégrale par rapport à celle de Riemann, et non pas de détailler la théorie sous-jacente. Le chapitre 3 expose les bases de la théorie des distributions, due à Laurent Schwartz, qui généralise la notion de fonction. Le chapitre 4 présente les propriétés fondamentales de la transformation de Fourier pour les fonctions intégrables et les fonctions de carré intégrable. Le chapitre 5 est consacré aux espaces de Hilbert qui, dans le cas de la dimension infinie, fournissent un cadre de travail analogue aux espaces euclidiens. Le chapitre 6 donne des exemples d’espaces de Hilbert qui jouent un rôle important dans l’étude des équations aux dérivées partielles : les espaces de Sobolev. Enfin, le chapitre 7 pose les bases de l’analyse variationnelle des problèmes elliptiques, qui ouvre notamment la porte à la méthode des éléments finis. Mais c’est là une autre histoire..
• New duality results for evenly convex optimization problems
  
  • Fajardo M.
  • Grad Sorin-Mihai
  • Vidal J.
  Optimization, Taylor & Francis, 2021, 70 (9), pp.1837-1858. (10.1080/02331934.2020.1756287)
  DOI : 10.1080/02331934.2020.1756287
• Personalized optimization with user’s feedback
  
  • Simonetto Andrea
  • Dall'Anese Emiliano
  • Monteil Julien
  • Bernstein Andrey
  Automatica, Elsevier, 2021, 131, pp.109767. (10.1016/j.automatica.2021.109767)
  DOI : 10.1016/j.automatica.2021.109767
• Complex-scaling method for the complex plasmonic resonances of planar subwavelength particles with corners
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  • Monteghetti Florian
  Journal of Computational Physics, Elsevier, 2021, 440. A subwavelength metallic particle supports localized surface plasmons for some negative permittivity values, which are eigenvalues of the self-adjoint quasi-static plasmonic eigenvalue problem (PEP). This work investigates the existence of complex plasmonic resonances for a 2D particle whose boundary is smooth except for one straight corner. These resonances are defined using the multivalued nature of some solutions of the corner dispersion relations and they are shown to be eigenvalues of a PEP that is complex-scaled at the corner, the finite element discretization of which yields a linear generalized eigenvalue problem. Numerical results show that the complex scaling deforms the essential spectrum (associated with the corner) so as to unveil both embedded plasmonic eigenvalues and complex plasmonic resonances. The later are analogous to complex scattering resonances with the local behavior at the corner playing the role of the behavior at infinity. These results corroborate the study of Li and Shipman (J. Integral Equ. Appl. 31(4), 2019), which proved the existence of embedded plasmonic eigenvalues and discussed the construction of particles that exhibit complex plasmonic resonances. (10.1016/j.jcp.2021.110433)
  DOI : 10.1016/j.jcp.2021.110433
• Solving Mixed Variational Inequalities Beyond Convexity
  
  • Grad Sorin-Mihai
  • Lara Felipe
  Journal of Optimization Theory and Applications, Springer Verlag, 2021, 190 (2), pp.565-580. Abstract We show that Malitsky’s recent Golden Ratio Algorithm for solving convex mixed variational inequalities can be employed in a certain nonconvex framework as well, making it probably the first iterative method in the literature for solving generalized convex mixed variational inequalities, and illustrate this result by numerical experiments. (10.1007/s10957-021-01860-9)
  DOI : 10.1007/s10957-021-01860-9
• Probabilistic backward McKean numerical methods for PDEs and one application to energy management.
  
  • Izydorczyk Lucas
  , 2021. This thesis concerns McKean Stochastic Differential Equations (SDEs) to representpossibly non-linear Partial Differential Equations (PDEs). Those depend not onlyon the time and position of a given particle, but also on its probability law. In particular, we treat the unusual case of Fokker-Planck type PDEs with prescribed final data. We discuss existence and uniqueness for those equations and provide a probabilistic representation in the form of McKean type equation, whose unique solution corresponds to the time-reversal dynamics of a diffusion process.We introduce the notion of fully backward representation of a semilinear PDE: thatconsists in fact in the coupling of a classical Backward SDE with an underlying processevolving backwardly in time. We also discuss an application to the representationof Hamilton-Jacobi-Bellman Equation (HJB) in stochastic control. Based on this, we propose a Monte-Carlo algorithm to solve some control problems which has advantages in terms of computational efficiency and memory whencompared to traditional forward-backward approaches. We apply this method in the context of demand side management problems occurring in power systems. Finally, we survey the use of generalized McKean SDEs to represent non-linear and non-conservative extensions of Fokker-Planck type PDEs.
• Propagation of elastic waves un buried waveguides: modelling of the forward problem and imaging with sampling methods
  
  • Fritsch Jean-François
  • Bourgeois Laurent
  • Hazard Christophe
  • Baronian Vahan
  • Recoquillay Arnaud
  , 2021.
• Distributed Multistage Optimization of Large-Scale Microgrids under Stochasticity
  
  • Pacaud François
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  IEEE Transactions on Power Systems, Institute of Electrical and Electronics Engineers, 2021. Microgrids are recognized as a relevant tool to absorb decentralized renewable energies in the energy mix. However, the sequential handling of multiple stochastic productions and demands, and of storage, make their management a delicate issue. We add another layer of complexity by considering microgrids where different buildings stand at the nodes of a network and are connected by the arcs; some buildings host local production and storage capabilities, and can exchange with others their energy surplus. We formulate the problem as a multistage stochastic optimization problem, corresponding to the minimization of the expected temporal sum of operational costs, while satisfying the energy demand at each node, for all time. The resulting mathematical problem has a large-scale nature, exhibiting both spatial and temporal couplings. However, the problem displays a network structure that makes it amenable to a mix of spatial decomposition-coordination with temporal decomposition methods. We conduct numerical simulations on microgrids of different sizes and topologies, with up to 48~nodes and 64~state variables. Decomposition methods are faster and provide more efficient policies than a state-of-the-art Stochastic Dual Dynamic Programming algorithm. Moreover, they scale almost linearly with the state dimension, making them a promising tool to address more complex microgrid optimal management problems.
• Multi-objective optimization for VM placement in homogeneous and heterogeneous cloud service provider data centers
  
  • Regaieg Rym
  • Koubàa Mohamed
  • Alès Zacharie
  • Aguili Taoufik
  Computing, Springer Verlag, 2021, 103 (6), pp.1255-1279. (10.1007/s00607-021-00915-z)
  DOI : 10.1007/s00607-021-00915-z
• Pseudo-compressibility, dispersive model and acoustic waves in shallow water flows
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Bristeau Marie-Odile
  • Godlewski Edwige
  • Imperiale Sébastien
  • Mangeney Anne
  • Sainte-Marie Jacques
  SEMA SIMAI Springer Series, Springer International Publishing, 2021, pp.209--250. In this paper we study a dispersive shallow water type model derived from the free surface compressible Navier-Stokes system. The compressible effects allow to capture the acoustic-like waves propagation and can be seen as a relaxation of an underlying incompressible model. The first interest of such a model is thus to capture both acoustic and water waves. The second interest lies in its numerical approximation. Indeed, at the discrete level, the pseudo-compressibility terms circumvent the resolution of an elliptic equation to capture the non-hydrostatic part of the pressure. This drastically reduces the cost of the numerical resolution of dispersive models especially in 2d and 3d. (10.1007/978-3-030-72850-2_10)
  DOI : 10.1007/978-3-030-72850-2_10
• Non overlapping Domain Decomposition Methods for Time Harmonic Wave Problems
  
  • Claeys Xavier
  • Collino Francis
  • Joly Patrick
  • Parolin Emile
  , 2021. The domain decomposition method (DDM) initially designed, with the celebrated paper of Schwarz in 1870 as a theoretical tool for partial differential equations (PDEs) has become, since the advent of the computer and parallel computing techniques, a major tool for the numerical solution of such PDEs, especially for large scale problems. Time harmonic wave problems offer a large spectrum of applications in various domains (acoustics, electromagnetics, geophysics, ...) and occupy a place of their own, that shines for instance through the existence of a natural (possibly small) length scale for the solutions: the wavelength. Numerical DDMs were first invented for elliptic type equations (e.g. the Laplace equation), and even though the governing equations of wave problems (e.g. the Helmholtz equation) look similar, standard approaches do not work in general.
• Non-local Impedance Operator for Non-overlapping DDM for the Helmholtz Equation
  
  • Collino Francis
  • Joly Patrick
  • Parolin Emile
  , 2021. In the context of time harmonic wave equations, the pioneering work of B. Després [4] has shown that it is mandatory to use impedance type transmission conditions in the coupling of sub-domains in order to obtain convergence of nonoverlapping domain decomposition methods (DDM). In later works [2, 3], it was observed that using non-local impedance operators leads to geometric convergence, a property which is unattainable with local operators. This result was recently extended to arbitrary geometric partitions, including configurations with cross-points, with provably uniform stability with respect to the discretization parameter [1]. We present a novel strategy to construct suitable non-local impedance operators that satisfy the theoretical requirements of [1] or [2, 3]. It is based on the solution of elliptic auxiliary problems posed in the vicinity of the transmission interfaces. The definition of the operators is generic, with simple adaptations to the acoustic or electromagnetic settings, even in the case of heterogeneous media. Besides, no complicated tuning of parameters is required to get efficiency. The implementation in practice is straightforward and applicable to sub-domains of arbitrary geometry, including ones with rough boundaries generated by automatic graph partitioners. We first provide in Section 1 a general definition of this novel transmission operator in a two-domain configuration. In Section 2 we then study more quantitatively the convergence in the geometric configuration of a closed wave-guide. Section 3 illustrates the results using actual finite element computations.
• Analysis of variational formulations and low-regularity solutions for time-harmonic electromagnetic problems in complex anisotropic media
  
  • Chicaud Damien
  • Ciarlet Patrick
  • Modave Axel
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2021, 53 (3), pp.2691-2717. We consider the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) is proven using well-suited functional spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analysed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H(curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark. (10.1137/20M1344111)
  DOI : 10.1137/20M1344111
• On learning node selection in a branch and bound algorithm
  
  • Alès Zacharie
  • Etheve Marc
  • Bissuel Côme
  • Juan Olivier
  • Kedad-Sidhoum Safia
  , 2021.
• Hamilton-Jacobi Approach for State-Constrained Differential Games and Numerical Learning Methods for Optimal Control Problems
  
  • Gammoudi Nidhal
  , 2021. This thesis will focus on the study of a theoretical and numerical approach for the multi-objective control problems with state constraints. Multi-objective optimization is an important approach for modelling complex problems in order to analyse the balance between different criteria to minimize. Here, the approach that will be used is based on the theory of Hamilton-Jacobi equations. The goal is to introduce a new methodology to study the properties and compute the Pareto front for multi-objective problems using the value function of an optimal control problem.
• Modélisation mathématique et méthode numérique pour la simulation du contrôle santé intégré par ultrasons de plaques composites stratifiées
  
  • Methenni Hajer
  , 2021. Ce sujet de thèse s’inscrit dans le contexte du contrôle intégré des structures, ou « Structural Health Monitoring » (SHM). Cette technique de contrôle non-destructif vise à utiliser un ou plusieurs capteurs, installés dans ou sur la structure d’intérêt. Le contrôle se fait in-situ et de façon périodique, afin d’obtenir des informations sur l’éventuelle apparition de défauts, tels que les défauts de corrosion pour les matériaux métalliques ou les défauts de délaminage pour les matériaux composites. Les données recueillies par les capteurs alimentent une analyse statistique dont le but est d’évaluer la santé de la structure au moment du contrôle, d’estimer son temps de vie restant et de faciliter la prise de décision quant à sa maintenance. Le SHM est de plus en plus présent dans de nombreux domaines industriels, en particulier dans le secteur aéronautique. Aussi le développement de modèles numériques pertinents comme performants constitue un atout majeur dans la conception de ces systèmes. Grâce à leur capacité à se propager sur de très grande distance, l’utilisation de capteurs ultrasonores générant des ondes guidées élastiques est une solution attirante. En pratique, des capteurs piézo-électriques fins, disposés à la surface de la structure, ou éventuellement enfouis pendant le procédé de fabrication, sont utilisés. Ils permettent l’émission et la réception des perturbations ultrasonores. Cependant, la nature dispersive des ondes guidées, combinée avec l’anisotropie inhérente aux matériaux composites rend difficile l’analyse des signaux obtenus lors du contrôle. De plus, proposer une modélisation fine de la propagation de ce type d’onde dans des configurations industrielles faisant intervenir des géométries complexes est une tâche difficile. Ces deux points constituent des obstacles non négligeables au développement de la méthodologie SHM, et l’objectif de cette thèse est de constituer l’ensemble des outils numériques qui permettront de proposer des solutions concrètes à ces problèmes.
• Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure
  
  • Boţ Radu Ioan
  • Grad Sorin-Mihai
  • Meier Dennis
  • Staudigl Mathias
  Advances in Nonlinear Analysis, De Gruyter, 2021, 10 (1), pp.450-476. Abstract In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems. (10.1515/anona-2020-0143)
  DOI : 10.1515/anona-2020-0143