  We are concerned with the reconstruction of a sound-soft obstacle using far
field measurements of the scattered waves associated with incident plane waves
sent from one direction but at multiple frequencies. We define, for each
frequency, the observable shape as the one which is described by finitely many
modes and produces a far field pattern close to the measured one. In the first
step, we propose a recursive Newton-type method for the reconstruction of the
observable shape at the highest frequency knowing an estimate of the observable
shape at the lowest frequency. We analyze its convergence and derive its
convergence rate in terms of the frequency step, the number of the Newton
iterations and the noise level. In the second step, we design a multilevel
Newton method which has the same convergence rate as the one described in the
first step but avoids the need of a good estimate of the observable shape at
the lowest frequency and a small frequency step (or a large number of Newton
iterations). The performances of the proposed algorithms are illustrated with
numerical results using simulated data.
