  17.  Topological Sensitivity for Solving Inverse Multiple Scattering Problems in 3D Electromagnetism. Part II : Iterative Method, F. Le Louër & M.L. Rapún, accepté pour publication dans SIAM J. Imaging Sci. (2018). abstract 
  
Abstract : In this work we study an iterative method based on the computation of iterated topological derivatives for the detection and shape identification of multiple electromagnetic scatterers. We derive closedform formulae for the topological derivative when an approximate set of domains has been already set. Either Neumann, Dirichlet, impedance or transmission conditions on the boundary of the scatterers are imposed. Proofs rely on the computation of shape derivatives followed by asymptotic expansions using Mie series derived from boundary integral formulations of the involved forward problems. Numerical results are included, illustrating the ability of the method to find shapes accurately without a priori information in a rather small number of iterations.

  16.  A Boundary Integral Equation for the Transmission Eigenvalue Problem for Maxwell's equation, F. Cakoni & O. Ivanyshyn & R. Kress & F. Le Louër, Math. Meth. Appl. Sci. 41 (2018), pp. 1316–1330. abstract 
  
Abstract : We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of the electrictomagnetic boundary trace operator which results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wavenumber. Further we employ the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn for the numerical computation of the transmission eigenvalues via this new integral equation.

  15.  A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles, F. Le Louër, ANZIAM eJournal 59 (2018), E1E49. abstract Fig. 2b Shape_and_Parameter 
  
Abstract : We consider fast and accurate solution methods for the direct and inverse scattering problems by a few three dimensional piecewise homogeneous dielectric obstacles around the resonance region. The forward problem is reduced to a system of second kind boundary integral equations. For the numerical solution of these coupled integral equations we modify a fast and accurate spectral algorithm, proposed by Ganesh and Hawkins [doi:10.1016/j.jcp.2008.01.016], by transporting these equations onto the unit sphere using the Piola transform of the boundary parametrisations. The computational performances of the forward solver are demonstrated on numerical examples for a variety of threedimensional smooth and non smooth obstacles. The algorithm, that requires the knowledge of the boundary parametrisation and leads to invert small linear systems, is wellsuited for the use of geometric optimisation tools to solve the inverse problem of recovering the shape of scatterers from the knowledge of noisy data. Computational details for the application of the iteratively regularised GaussNewton method to the numerical solution of the electromagnetic inverse problem are presented. Numerical experiments for the shape detection of multiple obstacles using incomplete radiation pattern data from back and front side are provided. The results in this article can also be applied for solving shape optimisation problems relying on timeharmonic electromagnetic waves.

  14.  Topological Sensitivity for Solving Inverse Multiple Scattering Problems in 3D Electromagnetism. Part I : One Step Method, F. Le Louër & M.L. Rapún, SIAM J. Imaging Sci. 10 (2017), n°3, pp. 12911321. abstract 
  
Abstract : In this paper we compute closedform expressions for the topological derivative for threedimensional timeharmonic electromagnetic waves for perfect conductors (Dirichlet condition), electromagnetic cavities (Neumann condition), absorbing obstacles (impedance condition), and dielectric inclusions (transmission conditions). The proofs are based on the computation of shape derivatives followed by asymptotic expansions using Mie series when infinitesimal spheres are considered. An exhaustive gallery of numerical experiments is presented, which demonstrate that the topological derivative is a very powerful tool for the detection of multiple electromagnetic scatterers without a priori information about their number, size, shape, or location. Numerical examples include highly demanding configurations where only a few incident directions and a few observation points (for nearfield data) or a few farfield observation directions (for farfield data) are considered.

  13.  Fast iterative BEM for highfrequency scattering problems in 3D elastodynamics, S. Chaillat & M. Darbas & F. Le Louër, Elsevier J. Comp. Phys. 341 (2017), pp. 429446. abstract 
  
Abstract : The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FMBEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FMBEM. The derivation of robust preconditioners for FMBEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FMBEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy
by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FMBEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [38]. The resulting boundary integral equations
are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

  12.  Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems, D. Kateb & F. Le Louër, Math. Mod. Meth. Appl. Sci. 26 (2016), n°10. abstract 
  
Abstract : This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three dimensional soundsoft or soundhard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called Generalised Impedance Boundary Condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first order GIBCs characterizing the presence of an interior thin layer with a constant thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non vanishing jumps across the exterior and the interior boundary of the thin layer. We show that we can interchange the first order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.

  11.  Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems, O. Ivanyshyn & F. Le Louër, Inverse Problems 32 (2016), n°9. abstract 
  
Abstract : This paper deals with the material derivative analysis of the boundary integral operators arising from the scattering theory of timeharmonic electromagnetic waves and its application to inverse problems. We present new results using the Piola transform of the boundary parametrisation to transport the integral operators on a fixed reference boundary. The transported integral operators are infinitely differentiable with respect to the parametrisations and simplified expressions of the material derivatives are obtained. Using these results, we extend a nonlinear integral equations approach developed for solving acoustic inverse obstacle scattering problems to electromagnetism. The inverse problem is formulated as a pair of nonlinear and illposed integral equations for the unknown boundary representing the boundary conditions and the measurements, for which the iteratively regularized GaussNewton method can be applied. The algorithm has the interesting feature that it avoids the numerous numerical solution of boundary value problems at each iteration step. Numerical experiments are presented in the special case of starshaped obstacles.

  10.  A domain derivativebased method for solving elastodynamic inverse obstacle scattering problems, F. Le Louër, Inverse Problems 31 (2015), n°11. abstract 
  
Abstract : The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from farfield data obtained by the scattering of a finite number of timeharmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Fréchet differentiability of the boundary to farfield operator and give a characterization of the first Fréchet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gauß Newton method and show numerical experiments in the special case of starshaped obstacles.

  9.  Approximate local DirichlettoNeumann map for threedimensional elastodynamic waves, S. Chaillat & M. Darbas & F. Le Louër, Comp. Meth. Appl. Mech. Eng. 217 (2015), pp. 6283. abstract 
  
Abstract : It has been proven that the knowledge of an accurate approximation of the DirichlettoNeumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for timeharmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic halfspace. These results are then extended to a general threedimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the OnSurface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies.

  8.  Wellconditioned boundary integral formulations for highfrequency elastic scattering problems in three dimensions, M. Darbas & F. Le Louër, Math. Meth. Appl. Sci. 38 (2015), pp. 1705–1733. abstract 
  
Abstract : We construct and analyze a family of wellconditioned boundary integral equations for the Krylov iterative solution of threedimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the wellknown BrakhageWerner and Combined Field Integral Equation formulations. We use a suitable approximation of the DirichlettoNeumann (DtN) map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate DtN map is inspired by the OnSurface Radiation Conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided.



7.

A
highorder spectral algorithm for elastic obstacle scattering in
three dimensions,
F. Le Louër, Elsevier J. Comp. Phys. 279 (2014), pp. 117. abstract 
  
Abstract : In this paper we describe a high order spectral algorithm for solving the timeharmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combinedfield boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developped by Ganesh and Hawkins  for solving second kind boundary integral equations in electromagnetism  to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parameterised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated on numerical examples for a variety of threedimensional convex and nonconvex smooth obstacles.

 
6.

Spectrally accurate numerical solution of hypersingular boundary integral equations for threedimensional electromagnetic wave scattering problems,
F. Le
Louër, Elsevier J. Comp. Phys. 275 (2014), pp. 662666. 


5.

On
the Fréchet derivative in elastic obstacle scattering,
F. Le Louër, SIAM J. Appl. Math., 72 (2012), n°5, pp. 1493–1507. abstract

  
Abstract : In this paper, we investigate the existence and characterizations of the Fréchet derivative of solutions to timeharmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique  the factorization of the difference of the farfield pattern for two different scatterers  introduced by Kress and Päivärinta to establish Fréchet differentiability in acoustic scattering. For the Dirichlet boundary condition an alternative proof of a differentiability result due to Charalambopoulos is provided and new results are proven for the Neumann and impedance exterior boundary value problems.



4.

Shape
derivatives of boundary integral operators in electromagnetic
scattering. Part II: Application to scattering by a homogeneous
dielectric obstacle,
M. Costabel
& F. Le Louër, Integr. Equ. Oper. Theory 73
(2012), n°1, pp. 1748. abstract

  
Abstract : We develop the shape derivative analysis of solutions to the problem of scattering of timeharmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity TH^{½}(div_{Γ},Γ). Using Helmholtz decomposition, we can base their analysis on the study of pseudodifferential integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.



3.

Shape
derivatives of boundary integral operators in electromagnetic
scattering. Part I: Shape differentiability of pseudohomogeneous
boundary integral operators,
M. Costabel
& F. Le Louër, Integr. Equ. Oper. Theory 72
(2012), n°4, pp. 509535. abstract 
  
Abstract : In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudohomogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.



2.

On
the use of Lamb modes in the linear sampling method for elastic
waveguides,
L. Bourgeois
& F. Le Louër & E.
Luneville, Inverse Problems 27
(2011), n°5, pp. 05500127. abstract

  
Abstract : This paper is devoted to a modal formulation of the linear sampling method in elastic 2D or 3D waveguide, that is we use the guided modes (called Lamb modes in 2D) as incident waves and the corresponding far fields in order to retrieve some obstacles. We provide the mathematical background to tackle the problem of identifiability and the justification of the linear sampling method for such a case. The elastic waveguide raises a specific issue: it concerns the projection of the scattered field on a transverse basis, which requires the introduction of new variables that mix displacement and stress components and satisfy the socalled Fraser's biorthogonality condition. Some numerical experiments in 2D show the feasibility of the reconstruction in the case of a finite number of incident waves formed by Lamb modes.



1.

On
the Kleinman  Martin integral equation method for electromagnetic
scattering by a dielectric body,
M. Costabel
& F. Le Louër, SIAM J. Appl. Math. 71
(2011), n°2, pp. 635656. abstract

  
Abstract : The interface problem describing the scattering of timeharmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface Γ. In this paper, following an idea developed by Kleinman and Martin for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over Γ. for a single unknown density. One knows that such boundary integral formulations of the Maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior Maxwell boundary value problem. We obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way, they become uniquely solvable for all real frequencies. We analyze the wellposedness of the integral equations in the space of finite energy on smooth and nonsmooth boundaries.
