Inverse Problems and Imaging doi:10.3934/ipi.2012.6.749 Volume 6, No. 4, 2012, 749–773 INVERSE ACOUSTIC OBSTACLE SCATTERING PROBLEMS USING MULTIFREQUENCY MEASUREMENTS Mourad Sini and Nguyen Trung Th` anh Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences Altenbergerstrasse 69, A-4040 Linz, Austria (Communicated by William Rundell) Abstract. In this paper, we investigate the problem of reconstructing sound- soft acoustic obstacles using multifrequency far field measurements correspond- ing to one direction of incidence. The idea is to obtain a rough estimate of the obstacle’s shape at the lowest frequency using the least-squares approach, then refine it using a recursive linearization algorithm at higher frequencies. Using this approach, we show that an accurate reconstruction can be obtained without requiring a good initial guess. The analysis is divided into three steps. Firstly, we give a quantitative estimate of the domain in which the least-squares objective functional, at the lowest frequency, has only one extreme (minimum) point. This result enables us to obtain a rough approximation of the obstacle at the lowest frequency from initial guesses in this domain using convergent gradient-based iterative procedures. Secondly, we describe the recursive lin- earization algorithm and analyze its convergence for noisy data. We qualita- tively explain the relationship between the noise level and the resolution limit of the reconstruction. Thirdly, we justify a conditional asymptotic H¨ older sta- bility estimate of the illuminated part of the obstacle at high frequencies. The performance of the algorithm is illustrated with numerical examples. 1. Introduction. The main objective of inverse obstacle scattering problems is to reconstruct the shape and/or physical properties of penetrable or impenetrable obstacles using the scattering theory of waves, for example acoustic, electromagnetic or elastic waves. Its applications can be found in several fields such as nondestructive evaluations, medical imaging, geophysics and remote sensing. In these problems, incident waves are sent and the waves scattered by the obstacles are measured either near (near field measurements) or far away (far field measurements) from the obstacles. In the literature, different methods have been used to solve the inverse scattering problems, depending on the types of measurements. For methods applied to measurements at a fixed frequency, we refer the reader to [6, 10, 20, 23] and the references therein. In this work, we consider the problem of reconstructing the shape of a sound-soft acoustic obstacle using far field measurements associated with incident plane waves sent from only one incident direction but at multiple frequencies. Using only one incident direction makes this type of measurements very feasible in practice, espe- cially when it is not possible or too time-consuming to perform the data acquisition for many different incident directions. The motivation for using multifrequency 2000 Mathematics Subject Classification. 35R30, 65N21, 78A46. Key words and phrases. Inverse obstacle scattering, multifrequency, recursive linearization al- gorithm, convergence, high frequency stability. 749 c 2012 American Institute of Mathematical Sciences