Monotonicity methods for inverse scattering problems

We consider two inverse scattering problems in unbounded free space. On the one hand, we investigate an inverse acoustic obstacle scattering problem governed by the time-harmonic Helmholtz equation. On the other hand, we examine an inverse electromagnetic medium scattering problem modeled by the time-harmonic Maxwell equations. In both cases, our goal is to recover the position and the shape of compactly supported scatterers D from far field observations of scattered waves. For the acoustic scattering problem, we assume that the scatterers are impenetrable obstacles that carry mixed Dirichlet and Neumann boundary conditions. For the electromagnetic scattering problem, the media are supposed to be penetrable, non-magnetic and non-absorbing but the electric permittivity may be inhomogeneous inside the scattering objects.
We approach both shape identification problems utilizing a monotonicity-based reconstruction ansatz. First, we establish monotonicity relations for the eigenvalues of the far field operators which map superpositions of plane wave incident fields to the far field patterns of the corresponding scattered fields. In addition, we discuss the existence of localized wave functions that have arbitrarily large energy in some prescribed region while at the same time having arbitrarily small energy in some other prescribed region. ... mehrCombining the monotonicity relations and the localized wave functions leads to rigorous characterizations of the support of the scattering objects. More precisely, we develop criteria that allow us to evaluate whether certain probing domains B are contained inside the unknown scatterer D or not and vice versa. Therefore, we introduce probing operators corresponding to the probing domains B and show that the number of positive or negative eigenvalues of suitable linear combinations of the far field operator corresponding to D and these probing operators is finite if and only if B is contained within D or if and only if B contains D. Finally, we complement our theoretical findings with numerical reconstruction algorithms and give some examples to illustrate the reconstruction procedure.