Publications

2013

• Numerical Microlocal analysis of 2-D noisy harmonic plane and circular waves
  
  • Benamou Jean-David
  • Collino Francis
  • Marmorat Simon
  Asymptotic Analysis, IOS Press, 2013, 83 (1-2), pp.157--187. We present a mathematical and numerical analysis of the stability and accuracy of the NMLA (Numerical MicroLocal Analysis) method [J. Comput. Phys. 199(2) (2004), 717-741] and its discretization.We restrict to homogeneous space and focus on the two simplest cases: (1) Noisy plane wave packets, (2) Noisy point source solutions. A stability result is obtained through the introduction of a new "impedance" observable. The analysis of the point source case leads to a modified second order (curvature dependent) correction of the algorithm. Since NMLA is local, this second order improved version can be applied to general data (heterogeneous media). See [J. Comput. Phys. 231(14) (2012), 4643-4661] for a an application to a source discovery inverse problem. (10.3233/ASY-121157)
  DOI : 10.3233/ASY-121157
• A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides
  
  • Fliss Sonia
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2013, 35 (2), pp.B438 - B461. This work deals with one-dimensional infinite perturbation---namely, line defects---in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. Contrary to existing methods, this one is exact, but there is a price to be paid: the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. © 2013, Society for Industrial and Applied Mathematics (10.1137/12086697X)
  DOI : 10.1137/12086697X
• Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: Mathematical justification of the Foldy-Lax model
  
  • Cassier Maxence
  • Hazard Christophe
  Wave Motion, Elsevier, 2013, 50 (1), pp.18-28. We are concerned with a two-dimensional problem which models the scattering of a time-harmonic acoustic wave by an arbitrary number of sound-soft circular obstacles. Assuming that their radii are small compared to the wavelength, we propose a mathematical justification of different levels of asymptotic models available in the physical literature, including the so-called Foldy-Lax model. © 2012 Elsevier B.V. (10.1016/j.wavemoti.2012.06.001)
  DOI : 10.1016/j.wavemoti.2012.06.001
• Analysis of the factorization method for a general class of boundary conditions
  
  • Chamaillard Mathieu
  • Chaulet Nicolas
  • Haddar Houssem
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013. We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the farfield operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the testing function. (10.1515/jip-2013-0013)
  DOI : 10.1515/jip-2013-0013
• Domain decomposition for the neutron SPN equations
  
  • Jamelot Erell
  • Ciarlet Patrick
  • Baudron Anne-Marie
  • Lautard Jean-Jacques
  , 2013. Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method applied to the neutron SPN equations, which are an approximation of the transport neutron equation. This method is based on the Schwarz iterative algorithm with optimized Robin interface conditions to handle communications. From a computational point of view, this method is rather easy to implement. We give some numerical results in highly heterogeneous 3D configurations. Computations are carried out with the MINOS solver, which is a multigroup SPN solver of the APOLLO3® neutronics code. Numerical experiments show that the method is robust and efficient, and that our choice of the Robin parameters is satisfactory.no abstract
• Plasmonic cavity modes: black-hole phenomena captured by Perfectly Matched Layers
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  , 2013. no abstract
• Scalar transmission problems between dielectrics and metamaterials: T-coercivity for the Discontinuous Galerkin approach.
  
  • Chung Eric T.
  • Ciarlet Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2013, 239, pp.189--207. no abstract
• On some expectation and derivative operators related to integral representations of random variables with respect to a PII process
  
  • Goutte Stéphane
  • Oudjane Nadia
  • Russo Francesco
  Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2013, 31, pp.108--141. Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and Föllmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance. (10.1080/07362994.2013.741395)
  DOI : 10.1080/07362994.2013.741395
• A general Hamilton-Jacobi framework for nonlinear state-constrained control problems
  
  • Altarovici Albert
  • Bokanowski Olivier
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (2), pp.337--357. The paper deals with deterministic optimal control problem with state constraints and non-linear dynamics. It is known for such a problem that the value function is in general discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumption, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypass the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach. (10.1051/cocv/2012011)
  DOI : 10.1051/cocv/2012011
• On the Well-Posedness , Stability And Accuracy Of An Asymptotic Model For Thin Periodic Interfaces In Electromagnetic Scattering Problems
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We analyze the well-posedness and stability properties of a parameter dependent problem that models the reflection and transmission of electromagnetic waves at a thin and rapidly oscillating interface. The latter is modeled using approximate interface conditions that can be derived using asymptotic expansion of the exact solution with respect to the small parameter (proportional to the periodicity length of oscillations and the width of the interface). The obtained uniform stability results are then used to analyze the accuracy (with respect to the small parameter) of the proposed model.
• Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Yu Tang Fei
  Journal of Computational Physics, Elsevier, 2013, 235, pp.14--31. In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell's equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee's scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency. (10.1016/j.jcp.2012.10.019)
  DOI : 10.1016/j.jcp.2012.10.019
• Obstacles in acoustic waveguides becoming "invisible" at given frequencies
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Nazarov Sergei
  Acoustical Physics / Akusticheskii zhurnal, MAIK Nauka/Interperiodica, 2013, 59(6), pp.633--639. We prove the existence of gently sloping perturbations of walls of an acoustic two-dimensional waveguide, for which several waves at given frequencies pass by the created obstacle without any distortion or with only a phase shift. (10.1134/S1063771013050047)
  DOI : 10.1134/S1063771013050047
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2013, pp.075012. Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods. (10.1088/0266-5611/29/7/075012)
  DOI : 10.1088/0266-5611/29/7/075012
• Stochastic analysis, random fields and applications VII
  
  • Russo Francesco
  • Dalang Robert C.
  • Dozzi Marco
  , 2013, 67, pp.xi + 469.
• On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems
  
  • Blanc F.
  • Touzé Cyril
  • Mercier Jean-François
  • Ege Kerem
  • Bonnet-Ben Dhia Anne-Sophie
  Mechanical Systems and Signal Processing, Elsevier, 2013, 36 (2), pp.520-539. Numerical computation of Nonlinear Normal Modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three di erent mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the center manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve. (10.1016/j.ymssp.2012.10.016)
  DOI : 10.1016/j.ymssp.2012.10.016
• Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces
  
  • Ciarlet Patrick
  Journal of Numerical Mathematics, De Gruyter, 2013, 21 (3), pp.173-180. Since it was originally designed, the Scott-Zhang interpolation operator has been very popular. Indeed, it possesses two keys features: it can be applied to fields without pointwise values and it preserves the boundary condition. However, no approximability properties seem to be available in the literature when the regularity of the field is weak. In this Note, we provide some estimates for such weakly regular fields, measured in Sobolev spaces with fractional order between 0 and 1 (10.1515/jnum-2013-0007)
  DOI : 10.1515/jnum-2013-0007
• Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
  
  • Jamelot Erell
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2013, 241, pp.445--463. no abstract