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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 Krasnoselskii-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., 30 pages, 21 figures

Intermittent sources of energy represent a challenge for electrical networks, particularly regarding demand satisfaction at peak times. Energy management tools such as load shaving or stora...

The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient $\alpha $ and compact terms. The coefficient $\alpha $ changes sign. The region where $\{\alpha>0\}$ is propagative, and the region where $\{\alpha

International audience; In bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations.

International audience; In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical $L^2$ framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle.

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., La simulation numérique de problèmes électromagnétiques dans des configurations physiques complexes est largement utilisée pour de nombreuses applications scientifiques et industrielles, telles que la conception de métamatériaux optiques ou l'étude des plasmas froids. L'analyse mathématique et numérique des problèmes de Maxwell est bien connue dans des contextes physiques simples, où les paramètres du milieu sont isotropes. Des résultats en milieux anisotropes existent, mais se limitent généralement au cas des tenseurs réels symétriques (ou complexes hermitiens) définis positifs. Cependant, pour certains milieux plus complexes, les problèmes ne sont pas couverts par la théorie standard. De nouveaux outils mathématiques doivent donc être développés pour analyser ces problèmes.Dans cette thèse, nous analysons des problèmes électromagnétiques harmoniques en temps pour une classe générale de tenseurs matériels anisotropes, appelés elliptiques. Nous développons un cadre fonctionnel étendu adapté à ces problèmes avec conditions limites de Dirichlet, Neumann ou Robin. Dans le cas de Robin, un intérêt particulier est porté à la caractérisation des espaces pour les traces de Robin. Nous étudions la régularité de la solution et de son rotationnel, et donnons des éléments d'analyse numérique. Dans la perspective de l'utilisation de méthodes de décomposition de domaine (DDM) pour une résolution accélérée, nous proposons et étudions différentes formulations décomposées, en nous focalisant sur leurs espaces fonctionnels et leur équivalence avec le problème global. Quelques expérimentations numériques sur la DDM complètent ce travail.

For minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in an Euclidean space we propose a stochastic incremental mirror descent algorithm constructed by means of the Nesterov smoothing. Further we modify the algorithm in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Next a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing is proposed in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions and a prox-friendly proper, convex and lower semicontinuous function. 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.

We are interested in path-dependent semilinear PDEs, where the derivatives are of Gâteaux type in specific directions [Formula: see text] and [Formula: see text], being the kernel functions of a Volterra Gaussian process [Formula: see text]. Under some conditions on [Formula: see text] and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is [Formula: see text].

International audience; It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which lead to suboptimal convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care.In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples.

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., Comment: 36 pages, 5 figures, 1 table

This paper is devoted to an acoustic scattering problem in a 2D partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a ‘bounded’ cross-section, while the right part is bounded in the transverse direction by some perfectly matched layers that mimic the situation of an open waveguide, that is with an ‘unbounded’ cross-section. We prove well-posedness of such scattering problem in the Fredholm sense (uniqueness implies existence) and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation.

Solution robustness focuses on structural similarities between the nominal solution and the scenario solutions. Most other robust optimization approaches focus on the quality robustness and only evaluate the relevance of their solutions through the objective function value. However, it can be more important to optimize the solution robustness and, once the uncertainty is revealed, find an alternative scenario solution $x^s$ which is as similar as possible to the nominal solution $x^{nom}$. This for example occurs when the robust solution is implemented on a regular basis or when the uncertainty is revealed late. We call this distance between $x^{nom}$ and $x^s$ the solution cost. We consider the proactive problem which minimizes the average solution cost over a discrete set of scenarios while ensuring the optimality of the nominal objective of $x^{nom}$. We show for two different solution distances $d_{val}$ and $d_{struct}$ that the proactive problem is NP-hard for both the integer min-cost flow problem with uncertain arc demands and for the integer max-flow problem with uncertain arc capacities. For these two problems, we prove that once the uncertainty is revealed, even identifying a reactive solution $x^r$ with a minimal distance to a given solution $x^{nom}$ is NP-hard for $d_{struct}$, and that it is polynomial for $d_{val}$. We highlight the benefits of solution robustness in a case study on a railroad planning problem. First, we compare our proactive approach to the anchored and the $k$-distance approaches. Then, we show the efficiency of the proactive solution over reactive solutions. Finally, we illustrate the solution cost reduction when relaxing the optimality constraint on the nominal objective of the proactive solution $x^{nom}$.

We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and $c$-subdifferentials are given. Formulae for the $c$-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality also by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework., Comment: 20 pages

International audience; 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.

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.

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International audience; In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set $\Omega$ of ${\mathbb R}^3$, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale $({\boldsymbol{H}}^{\mathtt{s}}(\Omega))_{\mathtt{s}>0}$. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived.

International audience; It is well known in the literature that standard hierarchical matrix (H-matrix) based methods, although very efficient for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous paper, we have shown that the method should nevertheless be used in the mechanical engineering community due to its still important data-compression rate and its straightforward implementation compared to H 2-matrix, or directional, approaches. Since in practice, not all materials are purely elastic it is important to be able to consider visco-elastic cases. In this context, we study the effect of the introduction of a complex wavenumber on the accuracy and efficiency of H-matrix based fast methods for solving dense linear systems arising from the discretization of the elastodynamic (and Helmholtz) Green's tensors. Interestingly, such configurations are also encountered in the context of the solution of transient purely elastic problems with the convolution quadrature method. Relying on the theory proposed in [12] for H 2-matrices for Helmholtz problems, we study the influence of the introduction of damping on the data compression rate of standard H-matrices. We propose an improvement of H-matrix based fast methods for this kind of configuration. This work is complementary to the recent work [12]. Here, in addition to addressing another physical problem, we consider standard H-matrices, derive a simple criterion to introduce additional compression and we perform extensive numerical experiments.

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., arXiv admin note: substantial text overlap with arXiv:1912.10902

We address the virtual machine placement problem that arises in Cloud Service Providers data centers. We purpose, a Multi-Objective Integer Linear Programming model which aims at optimizing simultaneously the number of hosted virtual machines, the resource wastage and the number of active physical machines (PM) in order to minimize power consumption. This new combination of objectives enables to maximize the client satisfaction rate with minimizing the data center (DC) operational costs. We modelize this problem with a multi-objective integer linear program and solve it through two different methods. The first method computes a unique solution for a given preference order over the objectives whereas the second computes a set of non-dominated solutions. Both methods are compared through extensive simulation scenarios. We consider two DC architectures: homogeneous DCs (i.e., a DC with PMs having the same amount of resources) and heterogeneous DCs. We study the impact of each DC configuration on the performances of the solutions. We show that the second method leads to solutions with a reduction of up to 30% over the number of used PMs and that the heterogeneous DCs outperforms the homogeneous one across all objectives.

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.

International audience; 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.

We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.

Many biological microswimmers locomote by periodically beating the densely-packed cilia on their cell surface in a wave-like fashion. While the swimming mechanisms of ciliated microswimmers have been extensively studied both from the analytical and the numerical point of view, the optimization of the ciliary motion of microswimmers has received limited attention, especially for non-spherical shapes. In this paper, using an envelope model for the microswimmer, we numerically optimize the ciliary motion of a ciliate with an arbitrary axisymmetric shape. The forward solutions are found using a fast boundary integral method, and the efficiency sensitivities are derived using an adjoint-based method. Our results show that a prolate microswimmer with a 2:1 aspect ratio shares similar optimal ciliary motion as the spherical microswimmer, yet the swimming efficiency can increase two-fold. More interestingly, the optimal ciliary motion of a concave microswimmer can be qualitatively different from that of the spherical microswimmer, and adding a constraint to the ciliary length is found to improve, on average, the efficiency for such swimmers., 19 pages, 7 figures

International audience; Inserting renewable energy in the electric grid in a decentralized manneris a key challenge of the energy transition. However, at local scale, both production and demand display erratic behavior, which makes it delicate to match them. It is the goal of Energy Management Systems (EMS) to achieve such balance at least cost. We present EMSx, a numerical benchmark for testing control algorithms for the management of electric microgrids equipped with a photovoltaic unit and an energy storage system. EMSx is made of three key components: the EMSx dataset, provided by Schneider Electric, contains a diverse pool of realistic microgrids with a rich collection of historical observations and forecasts; the EMSx mathematical framework is an explicit description of the assessment of electric microgrid control techniques and algorithms; the EMSx software EMSx.jl is a package, implemented in the Julia language, which enables to easily implement a microgrid controller and to test it. All components of the benchmark are publicly available, so that other researchers willing to test controllers on EMSx may reproduce experiments easily. Eventually, we showcase the results of standard microgrid control methods, including Model Predictive Control, Open Loop Feedback Control and Stochastic Dynamic Programming.

International audience; We consider the time harmonic scalar wave equation in junctions of several different periodic half-waveguides. In general this problem is not well posed. Several papers propose radiation conditions, i.e. the prescription of the behaviour of the solution at the infinities. This ensures uniqueness - except for a countable set of frequencies which correspond to the resonances- and yields existence when one is able to apply Fredholm alternative. This solution is called the outgoing solution. However, such radiation conditions are difficult to handle numerically. In this paper, we propose so-called transparent boundary conditions which enables us to characterize the outgoing solution. Moreover, the problem set in a bounded domain containing the junction with this transparent boundary conditions is of Fredholm type. These transparent boundary conditions are based on Dirichlet-to-Neumann operators whose construction is described in the paper. On contrary to the other approaches, the advantage of this approach is that a numerical method can be naturally derived in order to compute the outgoing solution. Numerical results illustrate and validate the method.

International audience; In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments.

The stochastic Auxiliary Problem Principle (APP) algorithm is a general Stochastic Approximation (SA) scheme that turns the resolution of an original optimization problem into the iterative resolution of a sequence of auxiliary problems. This framework has been introduced to design decomposition-coordination schemes but also encompasses many well-known SA algorithms such as stochastic gradient descent or stochastic mirror descent. We study the stochastic APP in the case where the iterates lie in a Banach space and we consider an additive error on the computation of the subgradient of the objective. In order to derive convergence results or efficiency estimates for a SA scheme, the iterates must be random variables. This is why we prove the measurability of the iterates of the stochastic APP algorithm. Then, we extend convergence results from the Hilbert space case to the Banach space case. Finally, we derive efficiency estimates for the function values taken at the averaged sequence of iterates or at the last iterate, the latter being obtained by adapting the concept of modified Fej{\'e}r monotonicity to our framework.

When we are interested in the propagation of waves in a periodic medium at low frequency (i.e. wavelength large compared to the period length), it is possible to model the periodic medium by an equivalent or effective homogeneous medium which has the same macroscopic properties. It is the homogenization theory that mathematically justifies this process. This process is very attractive because numerical calculations are then much less expensive (the small periodic structure has disappeared) and analytical calculations are again possible in certain configurations. The waves in the periodic medium and in the effective medium are very close from a macroscopic point of view except in the presence of boundaries or interfaces.Indeed, it is well known that the homogenized model is obtained by neglecting the boundary effects and consequently, it is much less precise at the boundaries of the periodic medium. When interesting phenomena appear at the edges of the middle (such as the propagation of plasmonic waves on the surface of metamaterials for example), it therefore seems difficult to trust the effective model.Returning to the homogenization process, we propose a homogenized model which is richer at the boundaries. The enriched homogenized model is as simple as the classical homogenized model far from the interfaces, only the boundary conditions change and take better account of the phenomena. We apply this model to an elliptical equation in the case of the simple geometry of the half-plane with Dirichlet or Neumann type conditions. From a numerical point of view, in addition to classic cell problems that appear in homogenization, periodic band problems must also be solved. Finally, we apply these results to the long time wave equation., Quand on s’intéresse à la propagation des ondes dans un milieu périodique à basse fréquence (i.e. la longueur d’onde est grande devant la période), il est possible de modéliser le milieu périodique par un milieu homogène équivalent ou effectif qui a les mêmes propriétés macroscopiques. C’est la théorie de l’homogénéisation qui justifie d’un point de vue mathématique ce procédé. Ce procédé est très séduisant car les calculs numériques sont beaucoup moins couteux (la petite structure périodique a disparu) et des calculs analytiques sont de nouveau possibles dans certaines configurations. Les ondes dans le milieu périodique et dans le milieu effectif sont très proches d’un point de vue macroscopique sauf en présence de bords ou d’interfaces.En effet, il est bien connu que le modèle homogénéisé est obtenu en négligeant les effets de bords et par conséquent il est beaucoup moins précis aux bords du milieu périodique. Quand les phénomènes intéressants apparaissent aux bords du milieu (comme la propagation des ondes plasmoniques à la surface des métamatériaux par exemple), il semble donc difficile de faire confiance au modèle effectif.En revenant sur le processus d’homogénéisation, nous proposons un modèle homogénéisé qui est plus riche aux niveaux des bords. Le modèle homogénéisé enrichi est aussi simple que le modèle homogénéisé classique loin des interfaces, seule les conditions aux bords changent et prennent mieux en compte les phénomènes. Nous appliquons ce modèle à une équation elliptique dans le cas de la géométrie simple du demi-plan avec des conditions de type Dirichlet ou Neumann. D’un point de vue numérique, en plus des problèmes de cellule classiques qui apparaissent en homogénéisation, des problèmes de bandes périodiques doivent également être résolus. Pour finir, nous appliquons ces résultats à l'homogénéisation de l'équation des ondes en temps long et en présence de bords.

Tsunami generation from earthquake induced seafloor deformations has long been recognized as a major hazard to coastal areas. Strike—slip faulting has generally been believed as insufficient for triggering large tsunamis, except through the generation of submarine landslides. Herein, we demonstrate that ground motions due to strike–slip earthquakes can contribute to the emergence of large tsunamis (>1m) underrather generic conditions. To this end, we have developed a computational framework that integrates models for earthquake rupture dynamics with models of tsunami generation and propagation. The three-dimensional time-dependent vertical and horizontal ground motions from spontaneous dynamic rupture models are used to drive boundary motions in the tsunami model. Our results suggest that super shearruptures propagating along strike–slip faults, traversing narrow and shallow bays are prime candidates for tsunami generation. We show that dynamic focusing and the large horizontal displacements, characteristic of strike-slip earthquakes on long faults, are critical drivers for the tsunami hazard. These findings point to intrinsic mechanisms for sizeable tsunami generation by strike–slip faulting, which do not require complex seismic sources, landslides, or complicated bathymetry. Furthermore, our model identifies three distinct phases in the tsunamic motion; an instantaneous dynamic phase, a lagging coseismic and a classical postseismic phase, each of which may affect coastal areas differently. We conclude that near-source tsunami hazards and risk from strike-slip faulting need to be re–evaluated.

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.

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., La thèse aura pour objectif d'étudier une approche théorique et numérique pour l'optimisation multi-objectif de trajectoires avec contraintes sur l’état. L'optimisation multi-objectif est une approche importante pour modéliser des problèmes complexes dans le but d’analyser le compromis entre différents critères à minimiser. Ici, l’approche qui sera utilisée est basée sur la théorie des équations de Hamilton-Jacobi. Le but est d’introduire une nouvelle méthodologie pour étudier les propriétés et calculer le front de Pareto pour les problèmes multi-objectif en utilisant la fonction valeur d’un problème de commande optimale.

International audience; Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein-Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle and regularity estimates on the solution.

H. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than $1$, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper arXiv:1801.04257[math.DG]., Comment: 18 pages

International audience; This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. {Among those, prolate spheroids were found to be the most efficient micro-swimmer shapes for a given reduced volume. We propose a simple shape-based scalar metric that can determine whether the optimal slip on a given shape makes it a pusher, a puller, or a neutral swimmer.}

International audience; The analysis of Cartesian Perfectly Matched Layers (PMLs) in the context of time-domain electromagnetic wave propagation in a 3D unbounded anisotropic homogeneous medium modelled by a diagonal dielectric tensor is presented. Contrary to the 3D scalar wave equation or 2D Maxwell's equations some diagonal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result.; Dans cette note nous nous intéressons à l’analyse de stabilité de la méthode de couches absorbantesparfaitement adaptées (PMLs) pour la propagation d’ondes électromagnétiques en régime transitoiredans un milieu anisotrope décrit par un tenseur diélectrique diagonal. Contrairement aux cas de l’équationd’ondes scalaire 3D et des équations de Maxwell 2D, certaines anisotropies diagonales mènent à l’existenced’ondes inverses qui provoquent des instabilités de la méthode PML. Ce résultat est illustré par des simulationsnumériques.

International audience; The well-posedness of the acoustic radiation in a 2D duct in presence of both a shear flow and an absorbing wall described by the Myers boundary condition is studied thanks to variational methods. Without flow the problem is found well-posed for any impedance value. The presence of a flow complicates the results. With a uniform flow the problem is proven to be always of the Fredholm type but is found well-posed only when considering a dissipative radiation problem. With a general shear flow, the Fredholm property is recovered for a weak enough shear and the dissipative radiation problem requires to introduce extra conditions to be well-posed: enough dissipation, a large enough frequency and non-intuitive conditions on the impedance value.

International audience; The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.

We study some effective transmission conditions able to reproduce the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits an acoustic Faraday cage able to resonate. In the study of Hewett & Hewitt (2016 Proc. R. Soc. A 472 , 20160062 ( doi:10.1098/rspa.2016.0062 )) different transmission conditions emerge from the asymptotic analysis whose validity depends on the frequency, specifically the distance to a resonance frequency of the cage. In practice, dealing with such conditions is difficult, especially if the problem is set in the time domain. In the present study, we demonstrate the validity of a simpler unified model derived in Marigo & Maurel (2016 Proc. R. Soc. A 472 , 20160068 ( doi:10.1098/rspa.2016.0068 )), where unified means valid whatever the distance to the resonance frequencies. The effectiveness of the model is discussed in the harmonic regime owing to explicit solutions. It is also exemplified in the time domain, where a formulation guaranteeing the stability of the numerical scheme has been implemented.

International audience; The concept of adaptive eigenspace basis (AEB) has recently proved effective for solving medium imaging problems. In this work, we present an AEB strategy for design parameterization in topology optimization (TO) problems. We seek the density design field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the structural domain, and solve for the associated eigenfunction coefficients. Restriction to this truncated eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution, removing the need for auxiliary filtering operations and design-variable bound constraints. We furthermore develop the basis adaptation scheme inherent in the AEB, which iteratively recomputes the eigenfunction basis to conform to the evolving density field, enabling further dimension reduction and acceleration of the optimization process. The known aptitude of the adapted eigenfunctions to approximate piecewise constant fields is especially useful for TO as relevant design subspaces can be given low-dimensional representations. We propose criteria for the selection of the basis dimension and demonstrate the use of basis function selection as means for length scale control. We compare performance of the AEB against conventional TO implementations in problems for static linear-elasticity, showing comparable structural solutions, computational cost benefits, and consistent design dimension reduction. K E Y W O R D S Adaptive eigenspace basis, Topology design, Dimensionality reduction Abbreviations: AEB, adaptive eigenspace basis; TO, topology optimization; EB, eigenspace basis. * Equally contributing authors.

We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number of qubits that scales logarithmically with the number of vertices of the graphs; and (2) on a new Ansatz that can efficiently probe the permutation space. Simulations are then presented to showcase the approach on graphs up to 16 vertices, whereas, given the logarithmic scaling, the approach could be applied to realistic sub-graph isomorphism problem instances in the medium term., Comment: 30 pages, 14 figures

International audience; When solving 2D linear elastodynamic equations in a homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated In [3, 2]. However in [4] the case of free surface boundary condition was proven to be unstable if a straight- forward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented in [4] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.

This thesis is embedded in the context of « Structural Health Monitoring ». This method of non-destructive testing aiming at monitoring in-situ an engineered structure is increasingly used in numerous industrial fields, e.g. the aeronautics industry. It is based upon elastic guided waves propagating over large distances. The interactions between incident wave fields and structural defects are gathered through a network of receiving sensors. The dispersive nature of guided waves, combined with the inherent anisotropy of some industrial materials, such as composites, makes the interpretation of the output signals difficult. The goal of this thesis is to provide meaningful numerical tools enhancing the understanding and analysis of propagation and interaction phenomena, appearing during the control experiment. The thesis lies between the physical and mathematical modelling of elastic waves and the construction of relevant numerical schemes, altogether in an innovating industrial context involving complex geometries and materials., 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.

In this paper, we study the $L^p$-asymptotic stability of the one-dimensional linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with $p\in (1,\infty)$. The damping term is assumed to be linear and localized to an arbitrary open sub-interval of $[0,1]$. We prove that the semi-group $(S_p(t))_{t\geq 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\geq 2$ or $1, Comment: 32 pages

International audience; Most industrial machines present vibrations. These vibrations can lead to premature aging, mechanical failure, local noise pollution or discomfort for people working nearby vibration sources. In particular, in the naval industry, vibrations can lead to several problems such as the deterioration of acoustic stealth for military applications, non-respect of the norms limiting the radiated noises on ships or the deterioration of underwater acoustic measurements on scientific ships. To offer efficient vibration damping solutions, PYTHEAS Technology has developed several devices based on piezoelectricity materials seeking to address the weaknesses of traditional solutions. Currently, the most commonly used solution consists of adding dissipative materials, often viscoelastic ones, which convert vibrations into heat. This solution has the advantage to be simple and cheap. However, its vibration damping performances are limited and its lifetime is often reduced due to creep. Active devices form the second main solution. They present better vibration damping performances, but their cost and complexity are obstacles to their deployment. PYTHEAS Technology has developed and tested several concepts of absorbers and suspensions which allow to reach a damping coefficient (ξ) equivalent to the one of active systems (ξ>10%),while remaining simple and having more than ten years of lifetime. Furthermore, the electromechanical conversion provided by the piezoelectric effect brings additional advantages : Vibration frequency tracking if it varies over time, Damping of several modes simultaneously using only one device (gain in terms of added mass and size), Device?s fine-tuning using the electronic (reduction of manufacturing costs),... This paper is an overview of the concepts developed by PYTHEAS Technology and their potential applications.

International audience; We consider an offshore wind farm defined by the location of the wind turbines and the amount of energy supplied per turbine, as well as the location of the central station responsible for redistributing the collected energy to users on the power grid. Knowing the power injected at each node, the capacities, susceptance and costs of the cables that can be used, the goal is to determine the least expensive cabling to route the energy supplied by the wind turbines to the central station. This cabling must respect the capacity constraints on the cables as well as the electrical constraints of Load Flow defined at each node of the network. In a second step, we look for a cabling that is also robust in case of failure of a cable, the notion of robustness being seen here as the protection again the worst case of failure. This work was carried out in collaboration with EDF Energies renouvelables. We give a mathematical model of the problem taking into account all the constraints of capacity, connectivity, load flow, cable types and incompatibility between edges, in the form of a mixed integer quadratic program that can be linearized and solved using a MIP solver. We then propose two mathematical models for the robust problem, formulation inspired by the previous one and a bi-level program where the second level is a max min program. Finally we present the results of our tests which provide solutions for real data up to about 50 nodes, before concluding.

This paper deals with a problem of trajectory optimization of the flight phases of a threestage launcher. The aim of this optimization problem is to minimize the consumption of ergols that is need to steer the launcher from the Earth to the GEO. Here we use a global optimization procedure based on Hamilton-Jacobi-Bellman (HJB) approach. An interesting by-product of the HJB approach is the synthesis of the optimal control in feedback form. Once the HJB equation is solved, for any starting point, the reconstruction of the optimal trajectory can be performed in real time. We aim at showing that combining several new techniques for the HJB approach we can obtain efficient solutions to a fully nonlinear control problem. The global optimization procedure proposed here takes also into account parametric optimisation that appears in the flight phases. Several recent advanced numerical techniques for HJB equations (high order finite difference schemes, parallel computing) are used to solve a 6-dimensional optimal control problem within a reasonable CPU time. This work has been carried out in the frame of CNES Launchers’ Research and Technology program