Convergence rates of recursive Newton-type methods for multifrequency scattering problems

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.