Publications

2019

• On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model
  
  • Bourgeois Laurent
  • Chesnel Lucas
  • Fliss Sonia
  Communications in Mathematical Sciences, International Press, 2019, 17 (6), pp.1487-1529. We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle. (10.4310/CMS.2019.v17.n6.a2)
  DOI : 10.4310/CMS.2019.v17.n6.a2
• Some contributions to the analysis of the Half-Space Matching Method for scattering problems and extension to 3D elastic plates
  
  • Tjandrawidjaja Yohanes
  , 2019. This thesis focuses on the Half-Space Matching Method which was developed to treat some scattering problems in complex infinite domains, when usual numerical methods are not applicable. In 2D, it consists in coupling several plane-wave representations in half-spaces surrounding the obstacle(s) with a Finite Element computation of the solution in a bounded domain. To ensure the matching of all these representations, the traces of the solution are linked by Fourier-integral equations set on the infinite boundaries of the half-spaces. In the case of a dissipative medium, this system of integral equations was proved to be coercive plus compact in an L² framework.In the present thesis, we derive error estimates with respect to the discretization parameters (both in space and Fourier variables). To handle the non-dissipative case, we propose a modified version of the Half-Space Matching Method, which is obtained by applying a complex-scaling to the unknowns, in order to recover the L² framework.We then extend the Half-Space Matching Method to scattering problems in infinite 3D elastic plates for applications to Non-Destructive Testing. The additional complexity compared to the 2D case comes from the decomposition on Lamb modes used in the half-plate representations. Due to the bi-orthogonality relation of Lamb modes, we have to consider as unknowns not only the displacement, but also the normal stress on the infinite bands limiting the half-plates. Some theoretical questions concerning this multi-unknown formulation involving the trace and the normal trace are studied in a 2D scalar case. Connections with integral methods are also addressed in the case where the Green's function is known, at least partially in each subdomain.The different versions of the method have been implemented in the library XLiFE++ and numerical results are presented for both 2D and 3D cases.
• Discrete-type approximations for non-Markovian optimal stopping problems: Part I
  
  • Leão Dorival
  • Ohashi Alberto
  • Russo Francesco
  Journal of Applied Probability, Cambridge University press, 2019, 56 (4), pp.981-1005. Abstract We present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al . (10.1017/jpr.2019.57)
  DOI : 10.1017/jpr.2019.57
• Injectivity of the inverse optimal control problem for control-affine systems
  
  • Jean Frédéric
  • Maslovskaya Sofya
  , 2019. Given a control system and a set of optimal trajectories, is it possible to recover the cost for which the trajectories are minimizing? This question is called inverse optimal control problem, and the problem is said to be injective when it admits a unique solution. In this paper we present a general approach to address the issue of the cost uniqueness in the class of quadratic costs and in the case of dynamics given by a control-affine system. We then apply this method to characterize the non-uniqueness cases for a special subclass of control-affine systems. (10.1109/CDC40024.2019.9028877)
  DOI : 10.1109/CDC40024.2019.9028877
• Embedded and high-order meshes : two alternatives to linear body-fitted meshes
  
  • Feuillet Rémi
  , 2019. The numerical simulation of complex physical phenomenons usually requires a mesh. In Computational Fluid Dynamics, it consists in representing an object inside a huge control volume. This object is then the subject of some physical study. In general, this object and its bounding box are represented by linear surface meshes and the intermediary zone is filled by a volume mesh. The aim of this thesis is to have a look on two different approaches for representing the object. The first approach called embedded method consist in integrally meshing the bounding box volume without explicitly meshing the object in it. In this case, the presence of the object is implicitly simulated by the CFD solver. The coupling of this method with linear mesh adaptation is in particular discussed.The second approach called high-order method consist on the contrary by increasing the polynomial order of the surface mesh of the object. The first step is therefore to generate a suitable high-order mesh and then to propagate the high-order information in the neighboring volume if necessary. In this context, it is mandatory to make sure that such modifications are valid and then the extension of classic mesh modification techniques has to be considered.
• Global optimization of polynomial programs with mixed-integer variables
  
  • Lazare Arnaud
  , 2019. In this thesis, we are interested in the study of polynomial programs, that is optimization problems for which the objective function and/or the constraints are expressed by multivariate polynomials. These problems have many practical applications and are currently actively studied. Different methods can be used to find either a global or a heuristic solution, using for instance, positive semi-definite relaxations as in the "Moment/Sum of squares" method. But these problems remain very difficult and only small instances are addressed. In the quadratic case, an effective exact solution approach was initially proposed in the QCR method. It is based on a quadratic convex reformulation, which is optimal in terms of continuous relaxation bound.One of the motivations of this thesis is to generalize this approach to the case of polynomial programs. In most of this manuscript, we study optimization problems with binary variables. We propose two families of convex reformulations for these problems: "direct" reformulations and quadratic ones.For direct reformulations, we first focus on linearizations. We introduce the concept of q-linearization, that is a linearization using q additional variables, and we compare the bounds obtained by continuous relaxation for different values of q. Then, we apply convex reformulation to the polynomial problem, by adding additional terms to the objective function, but without adding additional variables or constraints.The second family of convex reformulations aims at extending quadratic convex reformulation to the polynomial case. We propose several new alternative reformulations that we compare to existing methods on instances of the literature. In particular we present the algorithm PQCR to solve unconstrained binary polynomial problems. The PQCR method is able to solve several unsolved instances. In addition to numerical experiments, we also propose a theoretical study to compare the different quadratic reformulations of the literature and then apply a convex reformulation to them.Finally, we consider more general problems and we propose a method to compute convex relaxations for continuous problems.
• The Impact of Quadratization in Convexification-Based Resolution of Polynomial Binary Optimization
  
  • Lambert Amélie
  • Elloumi Sourour
  • Lazare Arnaud
  • Rodriguez-Heck Elisabeth
  , 2019.
• Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions
  
  • Monteghetti Florian
  • Haine Ghislain
  • Matignon Denis
  Mathematical Control and Related Fields, AIMS, 2019, 9 (4), pp.759-791. This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and Vũ (Studia Math., 88 (1988)). (10.3934/mcrf.2019049)
  DOI : 10.3934/mcrf.2019049
• Discrete-type approximations for non-Markovian optimal stopping problems: Part I
  
  • Leão Dorival
  • Ohashi Alberto
  • Russo Francesco
  Journal of Applied Probability, Cambridge University press, 2019, 56 (4), pp.981-1005. (10.1017/jpr.2019.57)
  DOI : 10.1017/jpr.2019.57
• Outils mathématiques et algorithmiques pour le calcul scientifique
  
  • Ciarlet Patrick
  • Jamelot Erell
  , 2019, pp.1-287. Ce polycopié correspond aux notes du cours "Calcul Scientifique Parallèle", tel qu'enseigné de 2014 à 2019 par les auteurs. Ce cours fait partie du cursus Modélisation et Simulation du M2 Analyse, Modélisation, Simulation de l'Université Paris-Saclay et du cursus de 3ème année ModSim de l'ENSTA Paris. L'objectif principal est de proposer aux étudiants des outils de calcul scientifique permettant d'appréhender les algorithmes adaptés au calcul parallèle, c'est-à-dire pouvant utiliser plusieurs nœuds de calcul simultanément. On abordera essentiellement le calcul parallèle d'un point de vue méthodologique et/ou algorithmique. A partir d'un problème modèle, on présente un certain nombre d'outils et de méthodes permettant de le résoudre numériquement, et on explique comment on peut adapter au calcul parallèle, c'est-à-dire paralléliser, les algorithmes associés. On évoque, sans les occulter, tous les aspects de la résolution, qu'ils soient abstraits (point de vue mathématique) ; discrétisation (point de vue numérique) ; algorithmique (mise en œuvre). A noter : le point de vue informatique du calcul parallèle est développé dans la documentation en ligne disponible à l'adresse https://ams301.pages.math.cnrs.fr/. Le polycopié est composé de trois parties et d'annexes. Dans la première partie, on rappelle quelques problèmes typiques à traiter. Parmi ces problèmes, on se concentrera sur la résolution de l'équation de diffusion des neutrons : résolution mathématique d'une part, et résolution numérique d'autre part. Pour ce second aspect, on introduit deux méthodes de discrétisation : les différences finies et les éléments finis. Les différences finies donnent lieu à des algorithmes de résolution numérique possédant une structure, on parle d'algorithmes structurés, alors que les éléments finis conduisent en général à des algorithmes non-structurés. Après discrétisation, l'opération fondamentale à réaliser est la résolution d'un système linéaire. La seconde partie se concentre donc sur l'algèbre linéaire numérique : éléments d'algorithmique numérique, les méthodes de résolution directes et itératives, les méthodes de Krylov et la méthode de la puissance itérée. La prise en compte de la structure, ou de l'absence de structure, joue un rôle déterminant dans la résolution parallèle. Enfin la troisième partie est une introduction aux méthodes de décomposition de domaine. Le calcul parallèle est naturellement associé à ces méthodes, car on choisit de découper le problème initial en plusieurs sous-problèmes interagissant entre eux, et on discrétise la seconde instance. On reprend comme exemple l'équation de diffusion des neutrons, discrétisée par la méthode des éléments finis. Après une introduction mathématique, on étudiera pour chaque problème deux méthodes de décomposition de domaine : la méthode de Schwarz et la méthode avec contrainte. On s'aidera de l'analyse numérique pour valider nos modèles décomposés. Les annexes comprennent des rappels en algèbre linéaire, les outils de base pour l'étude et l'approximation de formulations variationnelles en dimension infinie, et enfin quelques outils élémentaires sur les distributions et les espaces fonctionnels de type Sobolev.
• Coupled methods of nonlinear estimation and control applicable to terrain-aided navigation
  
  • Flayac Emilien
  , 2019. During this PhD, the general problem of designing coupled control and estimation methods for nonlinear dynamical systems has been investigated. The main target application was terrain-aided navigation (TAN), where the problem is to guide and estimate the 3D position of a drone flying over a known area. In this application, it is assumed that the only available data are the speed of the system, a measurement of the difference between the absolute altitude of the drone and the altitude of the ground flied over and a map of the ground. TAN is a good example of a nonlinear application where the separation principle cannot be applied. Actually, the quality of the observations depends on the control and more precisely on the area that is flied over by the drone. Therefore, there is a need for coupled estimation and control methods. It is to be noted that the estimation problem created by TAN is in itself difficult to analyse and solve. In particular, the following topics have been treated:• Nonlinear observer design and outputfeedback control for TAN with analytical ground mapsin a deterministic continuous-time framework.• The joint modelling of nonlinear optimal filtering and discrete-time stochastic optimal controlwith imperfect information.• The design of output-feedback Explicit dual stochastic MPC schemes coupled with a particlefilter and their numerical implementation to TAN.
• Energy Decay and Stability of a Perfectly Matched Layer For the Wave Equation
  
  • Baffet Daniel Henri
  • Grote Marcus J.
  • Imperiale Sébastien
  • Kachanovska Maryna
  Journal of Scientific Computing, Springer Verlag, 2019. In [25, 26], a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and L 2 stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations. Numerical results validate the theory. (10.1007/s10915-019-01089-9)
  DOI : 10.1007/s10915-019-01089-9
• Mathematical models for dispersive electromagnetic waves
  
  • Cassier Maxence
  • Joly Patrick
  • Kachanovska Maryna
  , 2019.
• A Unifying vision of Particle Filtering and Explicit Dual Control in Stochastic Control
  
  • Flayac Emilien
  • Dahia Karim
  • Hérissé Bruno
  • Jean Frédéric
  , 2019.
• Modeling multicrack propagation by the fast multipole symmetric Galerkin BEM
  
  • Dansou Anicet
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Engineering Analysis with Boundary Elements, Elsevier, 2019, 106, pp.309-319. The Fast Multipole Method coupled with the Symmetric Galerkin BEM is employed in this work to simulate fatigue crack growth. The resulted crack propagation code is accelerated with a fast matrix update, a parallel implementation and a sparse matrix format. By using multiple nodes, this code accommodates also multiple surface-breaking cracks. The numerical tests presented herein allow the propagation of multiple cracks in single or multilayer domains. (10.1016/j.enganabound.2019.05.019)
  DOI : 10.1016/j.enganabound.2019.05.019
• Invisible floating objects
  
  • Chesnel Lucas
  • Rihani Mahran
  , 2019. We consider a time-harmonic water waves problem in a 2D waveguide. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains two floating obstacles separated by a distance L. We study the behaviours of R (the reflection coefficient) and T (the transmission coefficient) as L tends to +∞. From this analysis, we exhibit situations of non reflectivity (R = 0, |T | = 1) or perfect invisibility (R = 0, T = 1). (10.34726/waves2019)
  DOI : 10.34726/waves2019
• Effective models for non-perfectly conducting thin coaxial cables
  
  • Beck Geoffrey
  • Imperiale Sebastien
  • Joly Patrick
  , 2019. Continuing past work on the modelling of coax-ial cables, we investigate the question of the modeling of non-perfectly conducting thin coax-ial cables. Starting from 3D Maxwell's equations , we derive, by asymptotic analysis with respect to the (small) transverse dimension of the cable, a simplified effective 1D model. This model involves a fractional time derivatives that accounts for the so-called skin effects in highly conducting regions.
• Recovering underlying graph for networks of 1D waveguides by reflectometry and transferometry
  
  • Beck Geoffrey
  • Bonnaud Maxime
  • Benoit Jaume
  , 2019. We present a method for blind recovery of network made out of a tree of 1D homogeneous waveguides with the same physical characteristics using reflectogram and transferogram(s).
• An efficient domain decomposition method with cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Antoine Xavier
  • Royer Anthony
  • Geuzaine Christophe
  , 2019. The parallel finite-element solution of large-scale time-harmonic scattering problems is addressed with a non-overlapping domain decomposition method (DDM). It is well known that the efficiency of this method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) are well suited for configurations without cross points (where more than two subdo-mains meet). In this work, we extend this approach to efficiently deal with cross points. Two-dimensional finite-element results are presented.
• Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces
  
  • Labarca Ignacio
  • Faria Luiz
  • Pérez-Arancibia Carlos
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2019, 475 (2227), pp.20190029. This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two spatial dimensions.The proposed methodology relies on convolution quadrature (CQ) schemes and the recently introduced windowed Green function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user’s choice is used to transform the problem into a finite number of (complex) frequency-domain problems posed, in our case, on the domains containing unbounded penetrable interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF metho—which introduces errors that decrease super-algebraically fast as the window size increases.The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Nyström or boundary element Helmholtz integral equation solvers capable of handling complex wavenumbers with large imaginary part. A high-order Nyström method based on Alpert’s quadraturerules is used here. A variety of CQ schemes and numerical examples, including wave propagation inopen waveguides as well as scattering from multiplelayered media, demonstrate the capabilities of the proposed approach. (10.1098/rspa.2019.0029)
  DOI : 10.1098/rspa.2019.0029
• Planewave Density Interpolation Methods for 3D Helmholtz Boundary Integral Equations
  
  • Pérez-Arancibia Carlos
  • Turc Catalin
  • Faria Luiz
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2019, 41 (4), pp.A2088-A2116. This paper introduce planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green’s third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nyström and boundary element discretization of boundary integral equation, can then be numerically evalu-ated by standard quadrature rules that are irrespective of the kernel singularity. Closed-formand purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nyström and Galerkin boundary element methods. A variety of numerical examples—including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique. (10.1137/19M1239866)
  DOI : 10.1137/19M1239866
• Reducing the Adaptation Costs of a Rolling Stock Schedule with Adaptive Solution: the Case of Demand Changes
  
  • Lucas Rémi
  • Alès Zacharie
  • Ramond François
  • Elloumi Sourour
  , 2019, 69, pp.857-876. In railway scheduling, a nominal traffic schedule is established well in advance for the main resources: train-paths, rolling stock and crew. However, it has to be adapted each time a change in the input data occurs. In this paper, we focus on the costs in the adaptation phase. We introduce the concept of adaptive nominal solution which minimizes adaptation costs with respect to a given set of potential changes. We illustrate this framework with the rolling stock scheduling problem with scenarios corresponding to increasing demand in terms of rolling stock units. We define adaptation costs for a rolling stock schedule and propose two MILPs. The first one adapts, at minimal cost, an existing rolling stock schedule with respect to a given scenario. The second MILP considers a set of given scenarios and computes an adaptive nominal rolling stock schedule together with an adapted solution to each scenario, again while minimizing adaptation costs. We illustrate our models with computational experiments on realistic SNCF instances.
• Impact of the Green function in acoustic analogies for flow noise predictions at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  , 2019. It is known that hydrodynamic noise can be a major contribution to the total sound radiated by a ship. It is in part attributed to the interaction between turbulent eddies with appendages and marine propeller blades. Because hydrodynamics is associated with very low Mach numbers, direct noise computation methods are too expensive. Other approaches must be chosen, based on acoustic analogies which consist first in modeling the incompressible turbulent flow and then in computing the noise radiated by this flow. We focus on Lighthill's wave equation, solved using the free space Green function or a tailored Green's function in presence of an arbitrary geometry. Unlike many studies from the literature where the impact of the chosen turbulent model is evaluated over a semi-infinite plate, the objective of this study is to evaluate the impact of the chosen Green function on the predicted broadband flow noise for a fixed semi-empirical turbulence model. The impact of the chosen tailored Green function on the radiated noise spectra and directivity diagrams is evaluated considering various analytical and numerical tailored Green's functions.
• Wave propagation in fractal trees. Mathematical and Numerical Issues
  
  • Joly Patrick
  • Kachanovska Maryna
  • Semin Adrien
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2019, 14 (2). We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments. (10.3934/nhm.2019010)
  DOI : 10.3934/nhm.2019010
• An efficient domain decomposition method with cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Antoine Xavier
  • Geuzaine Christophe
  , 2019. The parallel finite-element solution of large-scale time-harmonic scattering problems is addressed with a non-overlapping domain decomposition method. The efficiency of this method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Conditions based on high-order local absorbing boundary conditions have proved well suited for configurations without cross points (i.e. points where more than two subdomains meet). In this work, we extend this approach to efficiently deal with cross points. Two-dimensional numerical results are presented.