  The inverse wave scattering problem seeks to estimate a heterogeneous,
inaccessible medium, modeled by unknown variable coefficients in wave
equations, from transient recordings of waves generated by probing signals. It
is a widely studied inverse problem with important applications, that is
typically formulated as a nonlinear least squares data fit optimization. For
typical measurement setups and band-limited probing signals, the least squares
objective function has spurious local minima far and near the true solution, so
Newton-type optimization methods fail. We introduce a different approach, for
electromagnetic inverse wave scattering in lossless, anisotropic media. Our
reduced order model (ROM) is an algebraic, discrete time dynamical system
derived from Maxwell's equations with four important properties: (1) It is data
driven, without knowledge of the medium. (2) The data to ROM mapping is
nonlinear and yet the ROM can be obtained in a non-iterative fashion. (3) It
has a special algebraic structure that captures the causal Wave propagation.
(4) The ROM interpolates the data on a uniform time grid. We show how to obtain
from the ROM an estimate of the wave field at inaccessible points inside the
unknown medium. The use of this wave is twofold: First, it defines a
computationally inexpensive imaging function designed to estimate the support
of reflective structures in the medium, modeled by jump discontinuities of the
matrix valued dielectric permittivity. Second, it gives an objective function
for quantitative estimation of the dielectric permittivity, that has better
behavior than the least squares data fitting objective function. The
methodology introduced in this paper applies to Maxwell's equations in three
dimensions. To avoid high computational costs, we limit the study to a
cylindrical domain filled with an orthotropic medium, so the problem becomes
two dimensional.
