Inverse acoustic obstacle scattering problems using multifrequency measurements

Mourad Sini; Nguyen Trung Thành

doi:10.3934/ipi.2012.6.749

Article Contents

2012, Volume 6, Issue 4: 749-773. Doi: 10.3934/ipi.2012.6.749

This issue Previous Article Sampling type methods for an inverse waveguide problem Next Article

Inverse acoustic obstacle scattering problems using multifrequency measurements

Mourad Sini^1, and
Nguyen Trung Thành^1,

1.
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz

Received: July 31, 2011

Revised: July 31, 2012

Published: October 2012

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Abstract

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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Mathematics Subject Classification: 35R30, 65N21, 78A46.

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