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November  2012, 6(4): 749-773. doi: 10.3934/ipi.2012.6.749

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, Austria, Austria

Received  August 2011 Revised  August 2012 Published  November 2012

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.
Keywords: Inverse obstacle scattering, convergence, recursive linearization algorithm, high frequency stability., multifrequency.
Mathematics Subject Classification: 35R30, 65N21, 78A4.
Citation: Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749
References:
[1]

H.-D. Alber and A. G. Ramm, Scattering amplitude and algorithm for solving the inverse scattering problem for a class of nonconvex obstacles, J. Math. Anal. Appl., 117 (1986), 570-597.  Google Scholar

[2]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691 (electronic).  Google Scholar

[3]

H. Ammari, J. Garnier, H. Kang, M. Lim and K. SΦlna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564-600. Google Scholar

[4]

G. Bao and F. Triki, Error estimates for the recursive linearization of inverse medium problems, Journal of Computational Mathematics, 28 (2010), 725-744.  Google Scholar

[5]

O. Bucci, L. Crocco, T. Isernia and V. Pascazio, Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 1749-1756. Google Scholar

[6]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.  Google Scholar

[7]

Y. Chen, Inverse scattering via Heisenberg's uncertainty principle, Inverse Problems, 13 (1997), 253-282.  Google Scholar

[8]

J. Cheng and M. Yamamoto, Global uniqueness in the inverse acoustic scattering problem within polygonal obstacles, Chinese Ann. Math. Ser. B, 25 (2004), 1-6.  Google Scholar

[9]

W. Chew and J. Lin, A frequency-hopping approach for microwave imaging of large inhomogeneous bodies, IEEE Microwave and Guided Wave Letters, 5 (1995), 439-441. Google Scholar

[10]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.  Google Scholar

[11]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.  Google Scholar

[12]

G. B. Folland, "Fourier Analysis and its Applications," The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[13]

D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205.  Google Scholar

[14]

S. Gutman and A. G. Ramm, Support function method for inverse obstacle scattering problems, in "Acoustics, Mechanics, and the Related Topics of Mathematical Analysis," World Sci. Publ., River Edge, NJ, (2002), 179-184.  Google Scholar

[15]

F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587-620 (electronic).  Google Scholar

[16]

N. Honda, G. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Mathematische Annalen, appeared online 04 February, 2012. doi: 10.1007/s00208-012-0786-0.  Google Scholar

[17]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis," Reprint of the second (1990) edition, Classics in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar

[18]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.  Google Scholar

[19]

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96.  Google Scholar

[20]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[21]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares. Special section on imaging, Inverse Problems, 19 (2003), S91-S104.  Google Scholar

[22]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 2000.  Google Scholar

[23]

R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1-R47.  Google Scholar

[24]

A. G. Ramm, "Multidimensional Inverse Scattering Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 51, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992.  Google Scholar

[25]

E. Sincich and M. Sini, Local stability for soft obstacles by a single measurement, Inverse Probl. Imaging, 2 (2008), 301-315.  Google Scholar

[26]

P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc., 132 (2004), 1351-1354 (electronic).  Google Scholar

show all references

References:
[1]

H.-D. Alber and A. G. Ramm, Scattering amplitude and algorithm for solving the inverse scattering problem for a class of nonconvex obstacles, J. Math. Anal. Appl., 117 (1986), 570-597.  Google Scholar

[2]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691 (electronic).  Google Scholar

[3]

H. Ammari, J. Garnier, H. Kang, M. Lim and K. SΦlna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564-600. Google Scholar

[4]

G. Bao and F. Triki, Error estimates for the recursive linearization of inverse medium problems, Journal of Computational Mathematics, 28 (2010), 725-744.  Google Scholar

[5]

O. Bucci, L. Crocco, T. Isernia and V. Pascazio, Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 1749-1756. Google Scholar

[6]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.  Google Scholar

[7]

Y. Chen, Inverse scattering via Heisenberg's uncertainty principle, Inverse Problems, 13 (1997), 253-282.  Google Scholar

[8]

J. Cheng and M. Yamamoto, Global uniqueness in the inverse acoustic scattering problem within polygonal obstacles, Chinese Ann. Math. Ser. B, 25 (2004), 1-6.  Google Scholar

[9]

W. Chew and J. Lin, A frequency-hopping approach for microwave imaging of large inhomogeneous bodies, IEEE Microwave and Guided Wave Letters, 5 (1995), 439-441. Google Scholar

[10]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.  Google Scholar

[11]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.  Google Scholar

[12]

G. B. Folland, "Fourier Analysis and its Applications," The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[13]

D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205.  Google Scholar

[14]

S. Gutman and A. G. Ramm, Support function method for inverse obstacle scattering problems, in "Acoustics, Mechanics, and the Related Topics of Mathematical Analysis," World Sci. Publ., River Edge, NJ, (2002), 179-184.  Google Scholar

[15]

F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587-620 (electronic).  Google Scholar

[16]

N. Honda, G. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Mathematische Annalen, appeared online 04 February, 2012. doi: 10.1007/s00208-012-0786-0.  Google Scholar

[17]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis," Reprint of the second (1990) edition, Classics in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar

[18]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.  Google Scholar

[19]

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96.  Google Scholar

[20]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[21]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares. Special section on imaging, Inverse Problems, 19 (2003), S91-S104.  Google Scholar

[22]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 2000.  Google Scholar

[23]

R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1-R47.  Google Scholar

[24]

A. G. Ramm, "Multidimensional Inverse Scattering Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 51, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992.  Google Scholar

[25]

E. Sincich and M. Sini, Local stability for soft obstacles by a single measurement, Inverse Probl. Imaging, 2 (2008), 301-315.  Google Scholar

[26]

P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc., 132 (2004), 1351-1354 (electronic).  Google Scholar

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