Regularized recursive Newton-type methods for inverse scattering problems using multifrequency measurements^∗,∗∗

¹ Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria.
mourad.sini@oeaw.ac.at
² Department of Mathematics & Statistics, University of North Carolina at Charlotte, 9201 University City Blvd, Charlotte, NC 28223, USA. Current address: Department of Mathematics, Iowa State University, Carver Hall, Ames, IA 500011, USA.
thanh@iastate.edu

Received: 6 November 2013
Revised: 26 August 2014

Abstract

We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of scattered waves associated with incident plane waves sent from one incident direction but at multiple frequencies. We define, at each frequency, observable shapes as the ones which are described by finitely many modes and produce far field patterns close to the measured one. Our analysis consists of two steps. In the first step, we propose a regularized recursive Newton method for the reconstruction of an observable shape at the highest frequency knowing an estimate of an observable shape at the lowest frequency. We formulate conditions under which an error estimate in terms of the frequency step, the number of Newton iterations, and noise level can be proved. In the second step, we design a multilevel Newton method which has the same accuracy as the one described in the first step but with weaker assumptions on the quality of the 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.