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In this paper, we investigate the problem of reconstructing sound-soft acoustic obstacles using multifrequency far field measurements corresponding 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 linearization algorithm and analyze its convergence for noisy data. We qualitatively explain the relationship between the noise level and the resolution limit of the reconstruction. Thirdly, we justify a conditional asymptotic Hölder stability estimate of the illuminated part of the obstacle at high frequencies. The performance of the algorithm is illustrated with numerical examples.
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