12 research outputs found
Monotonicity in inverse scattering for Maxwell’s equations
We consider the inverse scattering problem to recover the support of penetrable scattering objects in three-dimensional free space from far field observations of scattered time-harmonic electromagnetic waves. The observed far field data are described by far field operators that map superpositions of plane wave incident fields to the far field patterns of the corresponding scattered waves. We discuss monotonicity relations for the eigenvalues of linear combinations of these operators with suitable probing operators. These monotonicity relations yield criteria and algorithms for reconstructing the support of scattering objects
from the corresponding far field operators. To establish these results we combine the monotonicity relations with certain localized vector wave functions that have arbitrarily large energy in some prescribed region while at the same time having arbitrarily small energy on some other prescribed region. Throughout we suppose that the relative magnetic permeability of the scattering objects is one, while their real-valued relative electric permittivity maybe inhomogeneous and the permittivity contrast may even change sign. Numerical examples
illustrate our theoretical findings
Monotonicity and local uniqueness for the Helmholtz equation
This work extends monotonicity-based methods in inverse problems to the case
of the Helmholtz (or stationary Schr\"odinger) equation in a bounded domain for fixed non-resonance frequency and real-valued
scattering coefficient function . We show a monotonicity relation between
the scattering coefficient and the local Neumann-Dirichlet operator that
holds up to finitely many eigenvalues. Combining this with the method of
localized potentials, or Runge approximation, adapted to the case where
finitely many constraints are present, we derive a constructive
monotonicity-based characterization of scatterers from partial boundary data.
We also obtain the local uniqueness result that two coefficient functions
and can be distinguished by partial boundary data if there is a
neighborhood of the boundary where and