American Institute of Mathematical Sciences

Advanced
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.   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.   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.   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.   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.   Google Scholar

[6]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar

[7]

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

[10]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Second edition, 93 (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.   Google Scholar

[12]

G. B. Folland, "Fourier Analysis and its Applications,", The Wadsworth & Brooks/Cole Mathematics Series, (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.   Google Scholar

[14]

S. Gutman and A. G. Ramm, Support function method for inverse obstacle scattering problems,, in, (2002), 179.   Google Scholar

[15]

F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems,, SIAM J. Numer. Anal., 37 (2000), 587.   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, (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, (1990).   Google Scholar

[18]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[19]

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

[22]

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

[23]

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

[24]

A. G. Ramm, "Multidimensional Inverse Scattering Problems,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 51 (1992).   Google Scholar

[25]

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

[6]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar

[7]

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

[10]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Second edition, 93 (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.   Google Scholar

[12]

G. B. Folland, "Fourier Analysis and its Applications,", The Wadsworth & Brooks/Cole Mathematics Series, (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.   Google Scholar

[14]

S. Gutman and A. G. Ramm, Support function method for inverse obstacle scattering problems,, in, (2002), 179.   Google Scholar

[15]

F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems,, SIAM J. Numer. Anal., 37 (2000), 587.   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, (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, (1990).   Google Scholar

[18]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[19]

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

[22]

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

[23]

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

[24]

A. G. Ramm, "Multidimensional Inverse Scattering Problems,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 51 (1992).   Google Scholar

[25]

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

[1]

Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems & Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77
[2]

Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026
[3]

Yuanjia Ma. The optimization algorithm for blind processing of high frequency signal of capacitive sensor. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1399-1412. doi: 10.3934/dcdss.2019096
[4]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[5]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[6]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[7]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[8]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131
[9]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039
[10]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[11]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[12]

Ali Fuat Alkaya, Dindar Oz. An optimal algorithm for the obstacle neutralization problem. Journal of Industrial & Management Optimization, 2017, 13 (2) : 835-856. doi: 10.3934/jimo.2016049
[13]

Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[14]

T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335
[15]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[16]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[17]

M. Montaz Ali. A recursive topographical differential evolution algorithm for potential energy minimization. Journal of Industrial & Management Optimization, 2010, 6 (1) : 29-46. doi: 10.3934/jimo.2010.6.29
[18]

Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems & Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537
[19]

Chun-Hsiung Hsia, Chang-Yeol Jung, Bongsuk Kwon. On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3319-3334. doi: 10.3934/dcdsb.2018322
[20]

Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences