Publications

2009

• Approximations of Stochastic Optimization Problems Subject to Measurability Constraints
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 19 (4), pp.1719-1734. Motivated by the numerical resolution of stochastic optimization problems subject to measurability constraints, we focus upon the issue of discretization. There exist indeed two components to be discretized for such problems, namely, the random variable modelling uncertainties (noise) and the $\sigma$-field modelling the knowledge (information) according to which decisions are taken. There is no reason to bind these two discretizations, which are a priori unrelated. In this setting, we present conditions under which the discretized problems converge to the original one. The focus is put on the convergence notions ensuring the quality of the approximation; we illustrate their importance by means of a counterexample based on the Monte Carlo approximation. Copyright © 2009 Society for Industrial and Applied Mathematics (10.1137/070692376)
  DOI : 10.1137/070692376
• Diffraction by a defect in an open waveguide: A Mathematical analysis based on a modal radiation condition
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Dakhia Ghania
  • Hazard Christophe
  • Chorfi Lahcène
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2009, 70 (3), pp.677-693. We consider the scattering of a time-harmonic acoustic wave by a defect in a twodimensional open waveguide. The scattered wave satisfies the Helmholtz equation in a perturbed layered half-plane. We introduce a modal radiation condition based on a generalized Fourier transform which diagonalizes the transverse contribution of the Helmholtz operator. The uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized Fourier variable. The existence is then deduced classically from Fredholm's alternative by reformulating the scattering problem as a Lippmann-Schwinger equation by means of the Green's function for the layered half-plane. © 2009 Society for Industrial and Applied Mathematics. (10.1137/080740155)
  DOI : 10.1137/080740155
• Control problems with mixed constraints and application to an optimal investment problem
  
  • Bonnans J. Frederic
  • Tiba Dan
  Mathematical Reports, Romanian Academy of Sciences, 2009, 4, pp.293-306.
• Comparison principle for a Generalized Fast Marching Method
  
  • Forcadel Nicolas
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.pp. 1923-1951. In \cite{CFFM06}, the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in \cite{CFFM06} for which we state a general comparison principle. We also prove the convergence of the new algorithm.
• Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements
  
  • Buffa Annalisa
  • Ciarlet Patrick
  • Jamelot Erell
  Numerische Mathematik, Springer Verlag, 2009, 113 (4), pp.497-518. A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with the help of a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, their method required a parameterization of the variational formulation. In order to avoid this difficulty, we use a mixed variational setting instead of the parameterization, which allows us to handle the divergence-free constraint on the field in a straightforward manner. The numerical analysis of the method is carried out, and numerical examples are provided to show the efficiency of our approach. © Springer-Verlag 2009. (10.1007/s00211-009-0246-2)
  DOI : 10.1007/s00211-009-0246-2
• Blockers and Transversals
  
  • Zenklusen Rico
  • Ries Bernard
  • Picouleau Christophe
  • de Werra Dominique
  • Costa Marie-Christine
  • Bentz Cédric
  Discrete Mathematics, Elsevier, 2009, 13, pp.4306--4314. We explore connections between d-blockers B in a graph G = (V;E) (i.e. subsets of edges whose removal decreases by at least d the cardinality of maximum matchings) and d-transversals T (i.e. subsets of edges such that every maximum matching M has at least d edges in T. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, grid graphs, chains and cycles. We also study the complexity status of finding minimum transversals and blockers. Algorithms for d-transversals and d- blockers based on dynamic programming are given for trees. (10.1016/j.disc.2009.01.006)
  DOI : 10.1016/j.disc.2009.01.006
• Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher
  
  • Martinon Pierre
  • Bonnans J. Frederic
  • Laurent-Varin Julien
  • Trélat Emmanuel
  Journal of Guidance, Control, and Dynamics, American Institute of Aeronautics and Astronautics, 2009, 32 (1), pp.51--55. We consider a flight mission to the geostationary transfer orbit (GTO) for an Ariane 5 launcher, while maximizing the payload or, as a variant, minimizing the fuel consumption. We first solve the complete flight sequence up to the final orbit, assuming a maximal thrust for all propulsion systems. Then we focus on the tmospheric ascent phase, which has been studied for instance in [1, 2, 3]. We are more specifically interested in optimal tra jectories with singular arcs (flight phases with a non maximal thrust) for the boosters. Due to the presence of tabulated data in the physical model, the exact expression of the singular control cannot be obtained from the time derivatives of the switching function. An alternate way to compute the singular control is provided, and numerical experiments are carried out for for several launcher variants.
• Stability Analysis of Optimal Control Problems with a Second-order State Constraint
  
  • Hermant Audrey
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 20 (1), pp.104-129. This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak second-order sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the $L^2$ norm, and Hölder continuous in the $L^\infty$ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 1998]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a ``primitive'' of the state constraint multiplier. (10.1137/070707993)
  DOI : 10.1137/070707993
• Direct computation of stresses in planar linearized elasticity
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (7), pp.1043-1064. Given a simply-connected domain Ω in ℝ2, consider a linearly elastic body with Ω as its reference configuration, and define the Hilbert space E(Ω)={e(eαβ) ∈ L2s (Ω) ∂11e22- 2∂12e12}+∂22e11 = 0 in H-2(Ω)}. Then we recently showed that the associated pure traction problem is equivalent to finding a 2 × 2 matrix field = (∈αβ) ∈E(Ω) that satisfies j(∈)= inf e∈E(Ω) j(e), where j(e) = 1/2 ∫Ω Aαβστ eστ eαβ dx - l(e), where (A αβστ ) is the elasticity tensor, and l is a continuous linear form over E(Ω) that takes into account the applied forces. Since the unknown stresses (σαβ) inside the elastic body are then given by σαβ = Aαβστ eστ, this minimization problem thus directly provides the stresses. We show here how the above Saint Venant compatibility condition ∂11 e22 - 2∂12e12 + ∂22e11 = 0 in H-2(Ω) can be exactly implemented in a finite element space h, which uses "edge" finite elements in the sense of J. C. Nédélec. We then establish that the unique solution h of the associated discrete problem, viz., find ∈h ∈ Eh such that j(∈h)=inf eh∈Eh j(eh) converges to in the space L2 s(Ω). We emphasize that, by contrast with a mixed method, only the approximate stresses are computed in this approach. © 2009 World Scientific Publishing Company. (10.1142/s0218202509003711)
  DOI : 10.1142/s0218202509003711
• Resonances of an elastic plate coupled with a compressible confined flow
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2009, 62 (2), pp.105-129. A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid-structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations λ(k) = k2, where λ is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically. © The author 2009. Published by Oxford University Press; all rights reserved. (10.1093/qjmam/hbp004)
  DOI : 10.1093/qjmam/hbp004
• Application of Discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic
  
  • Castel Nicolas
  • Cohen Gary
  • Duruflé Marc
  Journal of Computational Acoustics, World Scientific Publishing, 2009, 17 (2), pp.175-196. A discontinuous Galerkin method is developed for linear hyperbolic systems on general hexahedral meshes. The use of hexahedral elements and tensorized quadrature formulas to evaluate the integrals leads to an efficient matrix-vector product. It is shown for high order approximations, the reduction in computational time can be very important, compared to tetrahedral elements. Two choices of quadrature points are considered, the Gauss points or Gauss-Lobatto points. The method is applied to the aeroacoustic system (simplified Linearized Euler Equations). Some 3-D numericals experiments show the importance of penalization, and the advantage of using high order.
• Some convergence results for Howard's algorithm
  
  • Bokanowski Olivier
  • Maroso Stefania
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (4), pp.3001--3026. This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in \cA} (B^a x - b^a)=0$ where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and $\cA$ is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem. (10.1007/s00245-006-0865-2)
  DOI : 10.1007/s00245-006-0865-2
• Model reduction for a class of linear descriptor systems
  
  • Hechme Grace
  • Nechepurenko Yu.M.
  • Sadkane Miloud
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 229 (1), pp.54-60. For linear descriptor systems of the form Bẋ=Ax+Cu, this paper constructs reduced order systems associated with a given part of the finite spectrum of the pencil P(λ)=A−λB. It is known that the reduction can be obtained by a block diagonalization of the generalized Schur decomposition of P(λ). In this paper we consider the special case when B = [(H, 0; 0, 0)]and A = [(J, G; - F*, 0)]. This case is suited, in particular, for linearized hydrodynamic problems. We derive a sufficient condition under which the reduced system can approximate the initial one and show that it can be obtained in significantly cheap and efficient approaches. We consider first in detail the case when F = G and H is the identity matrix and then treat the general case. © 2008 Elsevier B.V. All rights reserved. (10.1016/j.cam.2008.10.001)
  DOI : 10.1016/j.cam.2008.10.001
• Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain.
  
  • Duruflé Marc
  • Grob Pascal
  • Joly Patrick
  Numerical Methods for Partial Differential Equations, Wiley, 2009, 25 (3), pp.526-551. In this paper, we examine the infl uence of numerical integration on finite element methods using quadrilateral or hexahedral meshes in the time domain. We pay special attention to the use of Gauss-Lobatto points to perform mass lumping for any element order. We provide some theoretical results through several error estimates that are completed by various numerical experiments. (10.1002/num.20353)
  DOI : 10.1002/num.20353
• Numerical approximation for a superreplication problem under gamma constraints
  
  • Bruder Benjamin
  • Bokanowski Olivier
  • Maroso Stefania
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.2289-2320. We study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized finite difference scheme is used since basic finite differences cannot work in our case. Numerical tests illustrate the validity of our approach. (10.1137/080725222)
  DOI : 10.1137/080725222
• Fast and accurate computation of layer heat potentials
  
  • Li Jing-Rebecca
  • Greengard Leslie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2009. We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as geometrically-induced stiffness, but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the history part of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.
• Numerical resolution of the wave equation on a network of slots
  
  • Semin Adrien
  , 2009, pp.35. In this technical report, we present a theoretical and numerical model to simulate wave propagation in finite networks of rods with both classical Kirchhoff conditions and Improved Kirchhoff conditions at the nodes of the networks. One starts with the continuous framework, then we discretize the problem using finite elements with the mass lumping technic introduced by G.~Cohen and P.~Joly. Finally, we show an implementation of the obtained numeric scheme in a homemade code written in C++ in collaboration with K.~Boxberger, some results and some error estimates.
• Revisiting the Analysis of Optimal Control Problems with Several State Constraints
  
  • Bonnans Joseph Frederic
  • Hermant Audrey
  Control and Cybernetics, Polish Academy of Sciences, 2009, 38 (4), pp.1021--1052.
• Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media
  
  • Fliss Sonia
  • Joly Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2009, 59 (9), pp.2155-2178. We consider the solution of the Helmholtz equation with absorption − u(x)−n(x)2(ω2 + ıε)u(x) = f (x), x = (x, y), in a 2D periodic medium Ω = R2. We assume that f (x) is supported in a bounded domain Ωi and that n(x) is periodic in the two directions in Ωe = Ω \ Ωi . We show how to obtain exact boundary conditions on the boundary of Ωi ,ΣS that will enable us to find the solution on Ωi . Then the solution can be extended in Ω in a straightforward manner from the values on ΣS . The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems. © 2008 IMACS. (10.1016/j.apnum.2008.12.013)
  DOI : 10.1016/j.apnum.2008.12.013
• Numerical analysis of the generalized Maxwell equations (with an elliptic correction) for charged particle simulations
  
  • Ciarlet Patrick
  • Labrunie Simon
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (11), pp.1959-1994. When computing numerical solutions to the Vlasov--Maxwell equations, the source terms in Maxwell's equations usually fail to satisfy the continuity equation. Since this condition is required for the well-posedness of Maxwell's equations, it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. These approaches, which involve a hyperbolic, a parabolic and an elliptic correction, have been recently analyzed mathematically. The goal of this paper is to carry out the numerical analysis for several variants of Maxwell's equations with an elliptic correction. (10.1142/S0218202509004017)
  DOI : 10.1142/S0218202509004017
• Space-time mesh refinement for discontunuous Galerkin methods for symmetric hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1886-1895. We present a new non-conforming space-time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method. (10.1016/j.cam.2009.08.094)
  DOI : 10.1016/j.cam.2009.08.094
• A survey on Polly Cracker Systems
  
  • Levy-Dit-Vehel Françoise
  • Marinari Maria-Grazia
  • Perret Ludovic
  • Traverso Carlo
  Gröbner Bases, Coding, and Cryptography, 2009.
• Existence of solutions for a model describing the dynamics of junctions between dislocations
  
  • Forcadel Nicolas
  • Monneau Régis
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2009, 40 (6), pp.pp. 2517-2535. We study a dynamical version of a multi-phase field model of Koslowski and Ortiz for planar dislocation networks. We consider a two-dimensional vector field which describes phase transitions between constant phases. Each phase transition corresponds to a dislocation line, and the vectorial field description allows the formation of junctions between dislocations. This vector field is assumed to satisfy a non-local vectorial Hamilton-Jacobi equation with non-zero viscosity. For this model, we prove the existence for all time of a weak solution. (10.1137/070710925)
  DOI : 10.1137/070710925
• Degree-constrained edge partitioning in graphs arising from discrete tomography
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Picouleau Christophe
  • Ries Bernard
  • de Werra Dominique
  Journal of Graph Algorithms and Applications, Brown University, 2009, 13 (2), pp.99-118. Starting from the basic problem of reconstructing a 2-dimensional im- age given by its projections on two axes, one associates a model of edge coloring in a complete bipartite graph. The complexity of the case with k = 3 colors is open. Variations and special cases are considered for the case k = 3 colors where the graph corresponding to the union of some color classes (for instance colors 1 and 2) has a given structure (tree, vertex- disjoint chains, 2-factor, etc.). We also study special cases corresponding to the search of 2 edge-disjoint chains or cycles going through specified vertices. A variation where the graph is oriented is also presented. In addition we explore similar problems for the case where the under- lying graph is a complete graph (instead of a complete bipartite graph). (10.7155/jgaa.00178)
  DOI : 10.7155/jgaa.00178
• The Aharonov-Bohm effect and Tonomura experiments: Rigorous results
  
  • Ballesteros Miguel
  • Weder Ricardo
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2009, 50 (12), pp.122108. The Aharonov-Bohm effect is a fundamental issue in physics. It describes the physically important electromagnetic quantities in quantum mechanics. Its experimental verification constitutes a test of the theory of quantum mechanics itself. The remarkable experiments of Tonomura ["Observation of Aharonov-Bohm effect by electron holography," Phys. Rev. Lett 48, 1443 (1982) and "Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave," Phys. Rev. Lett 56, 792 (1986)] are widely considered as the only experimental evidence of the physical existence of the Aharonov-Bohm effect. Here we give the first rigorous proof that the classical ansatz of Aharonov and Bohm of 1959 ["Significance of electromagnetic potentials in the quantum theory," Phys. Rev. 115, 485 (1959)], that was tested by Tonomura, is a good approximation to the exact solution to the Schrödinger equation. This also proves that the electron, that is, represented by the exact solution, is not accelerated, in agreement with the recent experiment of Caprez in 2007 ["Macroscopic test of the Aharonov-Bohm effect," Phys. Rev. Lett. 99, 210401 (2007)], that shows that the results of the Tonomura experiments can not be explained by the action of a force. Under the assumption that the incoming free electron is a Gaussian wave packet, we estimate the exact solution to the Schrödinger equation for all times. We provide a rigorous, quantitative error bound for the difference in norm between the exact solution and the Aharonov-Bohm Ansatz. Our bound is uniform in time. We also prove that on the Gaussian asymptotic state the scattering operator is given by a constant phase shift, up to a quantitative error bound that we provide. Our results show that for intermediate size electron wave packets, smaller than the ones used in the Tonomura experiments, quantum mechanics predicts the results observed by Tonomura with an error bound smaller than 10-99. It would be quite interesting to perform experiments with electron wave packets of intermediate size. Furthermore, we provide a physical interpretation of our error bound. © 2009 American Institute of Physics. (10.1063/1.3266176)
  DOI : 10.1063/1.3266176