Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

Abstract

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded 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 − 1 2 (divΓ, 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.

A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles

“…The characterisation of the derivative was then improved by Kress [34]. More recently, Fréchet differentiability was analyzed by Haddar and Kress [20] for the Neumann-impedance type obstacle scattering problem via the use of a far-field identity and by Costabel and Le Louër [9,10,36] and Hettlich [26] for the dielectric scattering problem via the boundary integral equation approach [8,36] and variational methods, respectively. This paper applies the iteratively regularised Gauss-Newton (irgn) method, combined with a fast forward solver, to solve the inverse scattering problem for multiple dielectric obstacles.…”

“…This transformation was first considered by Costabel and Le Louër [10] in the context of the shape differentiability analysis of the boundary integral operators M κ and C κ . However, for the numerical solution of boundary integral equations it is inconvenient as it requires explicit knowledge of the Hodge decomposition.…”

“…The following theorem was established by Costabel and Le Louër [10] for the scattering problem by a single dielectric obstacle. An alternative proof was proposed by Hettlich [26].…”

Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems

“…The computation of the iterates x requires the analysis and an explicit form of the first Fréchet derivative of the parametrized form  k of the boundary to far-field operator F k . The first Fréchet derivative is usually characterized as the far-field pattern of the solution to a new exterior boundary value problem [11,19,30]. As a consequence, the inverse scattering algorithm requires multiple numerical solution of boundary integral equations at each iteration step to compute the new iterates by solving a nonlinear least square problem via conjugate gradient method.…”

“…However, considering the integral operators being defined on the above mentioned energy space of mix regularity, the domain and the range of the transported operators still depend on the parametrizations. Costabel and Le Louër [11] get around this difficulty by exploiting the Helmholtz decomposition of the energy space [12]. In counterpart, one has to compute the material derivatives of a family of surface differential operators.…”