  We consider the reconstruction of the shape and the impedance function of an
obstacle from measurements of the scattered field at receivers outside the
object. The data is assumed to be generated by plane waves impinging on the
obstacle from multiple directions and at multiple frequencies. This inverse
problem is reformulated as the optimization problem of finding band-limited
shape and impedance functions which minimize the $L^2$ distance between the
computed value of the scattered field at the receivers and the data. The
optimization problem is non-linear, non-convex, and ill-posed. Moreover, the
objective function is computationally expensive to evaluate. The recursive
linearization approach (RLA) proposed by Chen has been successful in addressing
these issues in the context of recovering the sound speed of a domain or the
shape of a sound-soft obstacle. We present an extension of the RLA for the
recovery of both the shape and impedance functions. The RLA is a continuation
method in frequency where a sequence of single frequency inverse problems is
solved. At each higher frequency, one attempts to recover incrementally higher
resolution features using a step assumed to be small enough to ensure that the
initial guess obtained at the preceding frequency lies in the basin of
attraction for Newton's method at the new frequency. We demonstrate the
effectiveness of the method with several numerical examples. Surprisingly, we
find that one can recover the shape with high accuracy even when the
measurements are from sound-hard or sound-soft objects. While the method is
effective in obtaining high quality reconstructions for complicated geometries
and impedance functions, a number of interesting open questions remain. We
present numerical experiments that suggest underlying mechanisms of success and
failure, showing areas where improvements could help lead to robust and
automatic tools.
