Publications

2008

• An improved multimodal approach for non-uniform acoustic waveguides
  
  • Hazard Christophe
  • Lunéville Éric
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2008, 73 (4), pp.668-690. This paper explores from the point of view of numerical analysis a method for solving the acoustic time-harmonic wave equation in a locally non-uniform 2D waveguide. The multimodal method is based on a spectral description of the acoustic field in each transverse section of the guide, using Fourier-like series. In the case of sound-hard boundaries, the weak point of the method lies in the poor convergence of such series due to a poor approximation of the field near the boundaries. A remedy was proposed by Athanassoulis & Belibassakis (1999, J. Fluid Mech., 389, 275-301), where the convergence is improved thanks to the use of enhanced series. The present paper proposes a theoretical analysis of their idea and studies further improvements. For a general model which includes different kinds of non-uniformity (varying cross section, bend and inhomogeneous medium), a hybrid spectral/variational formulation is introduced. Error estimates are provided for a semi-discretized problem which concerns the effect of spectral truncation. These estimates are confirmed by numerical results. © The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. (10.1093/imamat/hxn006)
  DOI : 10.1093/imamat/hxn006
• A Fast Marching Method for Hamilton-Jacobi Equations Modeling Monotone Front Propagations
  
  • Cristiani Emiliano
  , 2008. In this paper we present a generalization of the Fast Marching method introduced by J. A. Sethian in 1996 to solve numerically the eikonal equation. The new method, named Buffered Fast Marching (BFM), is based on a semi-Lagrangian discretization and is suitable for Hamilton-Jacobi equations modeling monotonically advancing fronts, including Hamilton-Jacobi-Bellman and Hamilton-Jacobi- Isaacs equations which arise in the framework of optimal control problems and differential games. We also show the convergence of the algorithm to the viscosity solution. Finally we present several numerical tests proving that the BFM method is accurate and faster than the classical iterative algorithm in which every node of the grid is computed at every iteration.
• On a graph coloring problem arising from discrete tomography
  
  • Bentz Cédric
  • Costa Marie-Christine
  • de Werra Dominique
  • Picouleau Christophe
  • Ries Bernard
  Networks, Wiley, 2008, 51 (4), pp.256-267. Discrete tomography deals with the reconstruction of discrete homogenous objects from their projections. The reader is referred to the book of Hermann and Kuba [2] for an overview on discrete tomography. The image reconstruction problem is important since its solution is required for developing efficient procedures in image processing, data bases, crystallography, statistics, data compressing,... It can be formulated as follows: given a rectangular array where entries represent the pixels of a digitalized image coloured with k different colors, we consider the problem of reconstructing an image from the number of occurrences of each colour in every column and in every row. The problem is known to be polynomial for k=1, NP-complete for k=3 [1] and its complexity is still open for k=2. Here, we shall consider a graph colouring problem which generalizes both the well known basic graph colouring problem and the above image reconstruction problem. We are given a graph G=(V,E) and a collection P of p subsets Pi of vertices of G. We are also given a set of colours 1, 2, ..,k as well as a collection of p vectors h(Pi) of integers. The problem is to find a k-partition, i.e. a k-colouring, of V such that the number of vertices of Pi coloured with colour j is equal to the jth entry of the vector h(Pi), for all j=1,..,k and all i=1,..,p. The basic graph colouring problem deals with different colour assigned to adjacent vertices: we will call this a "proper" k-colouring otherwise we will call this simply a k-colouring. In this talk, we will consider colouring as well as proper colouring. Let us consider the special case where the graph G is a grid graph, the vertices, denoted by Xrs, are located on row r and column s, r=1,..,m and s=1,..,n, and P is the collection of the m+n chains corresponding to the rows and columns: the problem of finding a k-colouring of G corresponds exactly to the image reconstruction problem. In this talk we will consider some extensions by taking more general classes of graphs such as trees or bipartite graphs. We will restrict our attention to the case where each Pi is a chain in G. We call "cover index" of P, c(P), the maximum number of members of P which may cover a single element of V, i.e. which have a non empty intersection. We call "nested" a family P such that, for any pair of subsets, either one subset is included in the other or they are disjoint; then, the "nesticity" of P, nest(P), is the smallest number of nested families in a partition of P into nested families. First we will give several basic conditions for a solution to exist. Then, we will classify the problems according to the number of colours, the values of c(P) and nest(P), the class of the graph, the diameter of the graph, and so on. For each problem, either we will propose a polynomial time algorithm or we will give complexity results. For instance, we will prove that when G is a tree, the 2-colouring problem and the proper 3-colouring problems are NP-complete even if the maximum degree of G is bounded by 3; but we will propose a polynomial time algorithm solving the k-colouring problem when G is a tree and when any two Pi intersect in at most one vertex. All our results will be summarized in a table. (10.1002/net.20218)
  DOI : 10.1002/net.20218
• Conditional stability for ill-posed elliptic Cauchy problems : the case of Lipschitz domains (part II)
  
  • Bourgeois Laurent
  • Dardé Jérémi
  , 2008. This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with Lipschitz boundary. It completes the results obtained in \cite{bourgeois1} for domains of class $C^{1,1}$. This estimate is established by using an interior Carleman estimate and a technique based on a sequence of balls which approach the boundary. This technique is inspired from \cite{alessandrini}. We obtain a logarithmic stability estimate, the exponent of which is specified as a function of the boundary's singularity. Such stability estimate induces a convergence rate for the method of quasi-reversibility introduced in \cite{lions} to solve the Cauchy problems. The optimality of this convergence rate is tested numerically, precisely a discretized method of quasi-reversibility is performed by using a nonconforming finite element. The obtained results show very good agreement between theoretical and numerical convergence rates.
• Conditional stability for ill-posed elliptic Cauchy problems : the case of $C^{1,1}$ domains (part I)
  
  • Bourgeois Laurent
  , 2008. This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with $C^{1,1}$ boundary. It is an extension of an earlier result for domains of class $C^\infty$. Our estimate is established by using a global Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility to solve the ill-posed Cauchy problems.
• Higher order time stepping for second order hyperbolic problems and optimal CFL conditions
  
  • Gilbert Jean Charles
  • Joly Patrick
  , 2008, 16, pp.67-93. We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail and the analysis results in a specific numerical algorithm. The corresponding results are quite promising and suggest various conjectures. (10.1007/978-1-4020-8758-5_4)
  DOI : 10.1007/978-1-4020-8758-5_4
• Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2008. We present a new non conforming space-time mesh refinement method for 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.
• Vector and scalar potentials, Poincaré's theorem and Korn's inequality
  
  • Amrouche Chérif
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  , 2008. In this Note, we present several results concerning the vector potentials and the scalar potentials in a bounded, not necessarily simply-connected, three-dimensional domain. We consider also singular potentials corresponding to data in negative order Sobolev spaces. We also give some applications to Poincaré's theorem and to Korn's inequality.
• A new compactness result for electromagnetic waves. Application to the transmission problem between dielectrics and metamaterials
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  • Zwölf Carlo Maria
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2008, 18 (9), pp.1605-1631. We consider the time-harmonic Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic coefficients. We prove that, in the case of sign-shifting dielectric coefficients, the space of electric fields is compactly embedded in L 2. We build a three-field variational formulation equivalent to Maxwell system for sign-shifting magnetic coefficients and show that, under some suitable conditions, the formulation fits into the coercive plus compact framework. © 2008 World Scientific Publishing Company. (10.1142/s0218202508003145)
  DOI : 10.1142/s0218202508003145
• A spurious-free space-time mesh refinement for elastodynamics
  
  • Rodríguez Jerónimo
  International Journal for Multiscale Computational Engineering, Begell House, 2008, 6 (3), pp.263-279. We propose a generalization of the space-time mesh refinement technique for elastodynamics presented by 14 to the case where the discretization step (in space and time) on the fine grid is q N times finer than the one on the coarse grid. This method uses the conservation of a discrete energy to ensure the stability under the usual CFL condition. Some numerical examples show that the method is only first order accurate (and thus suboptimai with respect to the second-order interior scheme we have used) when the ratio of refinement is higher than 2. A Fourier analysis of the computed signals exhibits the presence of high-frequency waves (aliasing phenomena) polluting the fields on the fine grid. Those results provide valuable information with which to build a postprocessing by averaging that removes the spurious phenomena. Finally, we introduce a new numerical scheme, computing the postprocessed solution directly. This method is stable and second-order consistent, regardless of the ratio of refinement. Its performance is shown through a numerical simulation of the diffraction of elastic waves by small cracks. © 2008 by Begell House, Inc. (10.1615/intjmultcompeng.v6.i3.60)
  DOI : 10.1615/intjmultcompeng.v6.i3.60
• Spectral elements for the integral equations of time-harmonic Maxwell problems
  
  • Demaldent Édouard
  • Levadoux David
  • Cohen Gary
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2008, 56 (9), pp.3001-3010. We present a novel high-order method of moments (MoM) with interpolatory vector functions, on quadrilateral patches. The main advantage of this method is that the Hdiv conforming property is enforced, and at the same time it can be interpreted as a point-based scheme. We apply this method to field integral equations (FIEs) to solve time-harmonic electromagnetic scattering problems. Our approach is applied to the first and second Nédélec families of Hdiv conforming elements. It consists in a specific choice of the degrees of freedom (DOF), made in order to allow a fast integral evaluation. In this paper we describe these two sets of DOF and their corresponding quadrature rules. Sample numerical results on FIE confirm the good properties of our schemes: faster convergence rate and cheap matrix calculation. We also present observations on the choice of the discretization method, depending on the FIE selected. © 2008 IEEE. (10.1109/tap.2008.927551)
  DOI : 10.1109/tap.2008.927551
• The linear sampling method in a waveguide: A formulation based on modes
  
  • Bourgeois Laurent
  • Lunéville Éric
  Journal of Physics: Conference Series, IOP Science, 2008, 135 (-), pp.012023. This paper concerns the Linear Sampling Method to retrieve obstacles in a 2D or 3D acoustic waveguide. We derive a modal formulation of the LSM which is suitable for the waveguide configuration. Despite the ill-posedness of the inverse problem is increased owing to the evanescent modes, numerical experiments show good reconstruction of obstacles by using the far field. © 2008 IOP Publishing Ltd. (10.1088/1742-6596/135/1/012023)
  DOI : 10.1088/1742-6596/135/1/012023