Analysis of the factorization method for a general class of boundary conditions

Abstract

SubmittedInternational audienceWe 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

Analysis of new direct sampling indicators for far-field measurements

“…However, one can still apply the FM either to the operators Im(F ) or F = |Re(F )| + |Im(F )| (see for e.g. [8,17,18,21]). By definition one has that these operators are self-adjoint and compact which implies they have an orthonormal spectral decomposition.…”

“…Here we propose approximating the solution operator using Tikhonov regularization, which is commonly used in the literature to solve (6) (see for e.g. [8]).…”

The electromagnetic scattering problem with generalized impedance boundary conditions

“…First of all, uniqueness holds from Theorem 3.4. To prove existence we adapt the procedure presented in [12, chapter 10] in the case of a Dirichlet type boundary condition to the volume formulation (6). We do not give a precise proof but we only highlight the main steps since it is rather classical.…”

“…Driven by recent advances in the study of inverse acoustic scattering problems in the presence of so-called generalized impedance boundary conditions (see [2,3,4,6]) we study in this paper well-posedness of the forward electromagnetic scattering problem in the harmonic regime in the case where the scatterer is characterised by a boundary condition of the form ν × E + ZH T = f on Γ where Γ is the boundary of the scatterer, ν is the outward unit normal vector to Γ, E is the electric field, H T stands for the tangential component of the magnetic field H, Z is a surface differential operator and f is a source term. This kind of boundary conditions, often referred to as Generalized Impedance Boundary Condition, are known to provide accurate models for all sort of small scale structures.…”

“…Nevertheless, with these standard approach one needs to assume some compatibility between the signs of the surface operator Z and the sign of the volume contribution to the variational formulation to ensure reasonable coercivity properties. Actually, at least for the acoustic scattering problem problem (see [6,13]), it seems that such restrictive conditions are not needed. To clarify this point, we consider a different formulation for the scattering problem which consists in writing the problem as a single operator equation posed on the boundary of the scatterer.…”