Skip to main content
SpringerLink
Log in

Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at a collection of receivers outside the object. The data is assumed to be generated by plane waves impinging on the unknown obstacle from multiple directions and at multiple frequencies. This inverse problem can be reformulated as an optimization problem: that of finding band-limited shape and impedance functions which minimize the L2 distance between the computed value of the scattered field at the receivers and the given measurement data. The optimization problem is highly non-linear, non-convex, and ill-posed. Moreover, the objective function is computationally expensive to evaluate (since a large number of Helmholtz boundary value problems need to be solved at every iteration in the optimization loop). The recursive linearization approach (RLA) proposed by Chen has been successful in addressing these issues in the context of recovering the sound speed of an inhomogeneous object or the shape of a sound-soft obstacle. We present an extension of the RLA for the recovery of both the shape and impedance functions of the object. The RLA is, in essence, a continuation method in frequency where a sequence of single frequency inverse problems is solved. At each higher frequency, one attempts to recover incrementally higher resolution features using a step assumed to be small enough to ensure that the initial guess obtained at the preceding frequency lies in the basin of attraction for Newton’s method at the new frequency. We demonstrate the effectiveness of this approach with several numerical examples. Surprisingly, we find that one can recover the shape with high accuracy even when the measurements are generated by sound-hard or sound-soft objects, eliminating the need to know the precise boundary conditions appropriate for modeling the object under consideration. While the method is effective in obtaining high-quality reconstructions for many complicated geometries and impedance functions, a number of interesting open questions remain regarding the convergence behavior of the approach. We present numerical experiments that suggest underlying mechanisms of success and failure, pointing out areas where improvements could help lead to robust and automatic tools for the solution of inverse obstacle scattering problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alpert, B.K.: Hybrid gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20, 1551–1584 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aslanyürek, B., Haddar, H., Şahintürk, H.: Generalized impedance boundary conditions for thin dielectric coatings with variable thickness. Wave Motion 48, 681–700 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aslanyürek, B., Sahintürk, H.: Reconstruction of thickness variation of a dielectric coating through the generalized impedance boundary conditions. ESAIM Math. Model. Numer. Anal. 48, 1011–1027 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, G., Hou, S., Li, P.: Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm. J. Comput. Phys. 227, 755–762 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao, G., Li, P.: Inverse medium scattering problems for electromagnetic waves. SIAM J. Appl. Math. 65, 2049–2066 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, G., Li, P., Lin, J., Triki, F.: Inverse scattering problems with multi-frequencies. Inverse Probl. 31, 093001 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bao, G., Lin, J.: Imaging of local surface displacement on an infinite ground plane: the multiple frequency case. SIAM J. Appl. Math. 71, 1733–1752 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bao, G., Lin, J., Triki, F.: A multi-frequency inverse source problem. J. Differ. Equ. 249, 3443–3465 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bao, G., Lin, J., Triki, F., et al.: Numerical solution of the inverse source problem for the helmholtz equation with multiple frequency data. Contemp. Math. 548, 45–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bao, G., Lu, S., Rundell, W., Xu, B.: A recursive algorithm for multifrequency acoustic inverse source problems. SIAM J. Numer. Anal. 53, 1608–1628 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bao, G., Triki, F.: Error estimates for the recursive linearization of inverse medium problems. J. Comput. Math., 725–744 (2010)

  12. Bebendorf, M.: Hierarchical LU decomposition-based preconditioners for BEM. Computing 74, 225–247 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Beylkin, D., Rokhlin, V.: Fitting a bandlimited curve to points in a plane. SIAM J. Sci. Comput. 36, A1048–A1070 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Borges, C., Biros, G.: Reconstruction of a compactly supported sound profile in the presence of a random background medium. Inverse Probl. 34, 115007 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Borges, C., Gillman, A., Greengard, L.: High resolution inverse scattering in two dimensions using recursive linearization. SIAM J Imaging Sci. 10, 641–664 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Borges, C., Greengard, L.: Inverse obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence. SIAM J. Imaging Sci. 8, 280–298 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Borges, C., Lai, J.: Inverse scattering reconstruction of a three dimensional sound-soft axis-symmetric impenetrable object. Inverse Probl. 36, 105005 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices. Lect. Notes 21, 2003 (2003)

    MATH  Google Scholar 

  19. Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Eng. Anal. Bound. Elem. 27, 405–422 (2003)

    Article  MATH  Google Scholar 

  20. Bourgeois, L., Chaulet, N., Haddar, H.: Stable reconstruction of generalized impedance boundary conditions. Inverse Probl. 27, 095002 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bourgeois, L., Chaulet, N., Haddar, H.: On simultaneous identification of the shape and generalized impedance boundary condition in obstacle scattering. SIAM J. Sci. Comput. 34, A1824–A1848 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bourgeois, L., Haddar, H.: Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Probl. Imaging 4, 19–38 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bruno, O., Elling, T., Turc, C.: Regularized integral equations and fast high-order solvers for sound-hard acoustic scattering problems. Int. J. Num. Meth. Eng. 91, 1045–1072 (2012)

    Article  MathSciNet  Google Scholar 

  24. Cakoni, F., Kress, R.: Integral equation methods for the inverse obstacle problem with generalized impedance boundary condition. Inverse Probl. 29, 015005 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chaillat, S., Biros, G.: FaIMS: a fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar Helmholtz equation. J. Comput. Phys. 231, 4403–4421 (2012)

    Article  MATH  Google Scholar 

  26. Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 29, 67–81 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chandrasekaran, S., Gu, M., Pals, T.: A fast ULV decomposition solver for hierarchically semiseparable representations. SIAM J. Matrix Anal. Appl. 28, 603–622 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chavent, G., Papanicolaou, G., Sacks, P., Symes, W.: Inverse Problems in Wave Propagation, The IMA Volumes in Mathematics and its Applications. Springer, New York (2012)

    Google Scholar 

  29. Chen, Y.: Recursive linearization for inverse scattering, Mathematical and numerical aspects of wave propagation (Golden CO, 1998), pp. 114–117 (1995)

  30. Chen, Y.: Inverse scattering via Heisenberg’s uncertainty principle. Inverse Probl. 13, 253 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Cheney, M., Borden, B.: Fundamentals of radar imaging, CBMS-NSF Regional Conference Series in Applied Mathematics Society for Industrial and Applied Mathematics (2009)

  32. Collins, M.D., Kuperman, W.A.: Inverse problems in ocean acoustics. Inverse Probl. 10, 1023 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. Collins, R.: Nondestructive Testing of Materials, Studies in Applied Electromagnetics and Mechanics. IOS Press, Amsterdam (1995)

    Google Scholar 

  34. Colton, D., Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Probl. 12, 383 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Colton, D., Kress, R.: Integral equation methods in scattering theory. SIAM (2013)

  36. Colton, DL, Kress, R: Inverse acoustic and electromagnetic scattering theory, 4th edn. Springer, New York (2019)

    Book  MATH  Google Scholar 

  37. Engl, H., Louis, A., Rundell, W.: Inverse problems in medical imaging and nondestructive testing: proceedings of the conference in Oberwolfach, Federal Republic of Germany, February 4–10, 1996. Springer, Vienna (2012)

    MATH  Google Scholar 

  38. Farhat, C., Tezaur, R., Djellouli, R.: On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method. Inverse Probl. 18, 1229–1246 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gillman, A., Young, P.M., Martinsson, P.G.: A direct solver with \(\mathcal {O}({N})\) complexity for integral equations on one-dimensional domains. Front. Math. China 7, 217–247 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Greengard, L., Gueyffier, D., Martinsson, P.G., Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numerica 18, 243–275 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Guo, J., Yan, G., Cai, M.: Multilayered scattering problem with generalized impedance boundary condition on the core. J. Appl. Math., 2015 (2015)

  42. Gutman, S., Klibanov, M.: Regularized quasi-Newton method for inverse scattering problems. Math. Comput. Model. 18, 5–31 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  43. Haddar, H., Joly, P., Nguyen, H.M.: Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Models Methods Appl. Sci. 15, 1273–1300 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Haddar, H., Joly, P., Nguyen, H.M.: Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell’s equations. Math. Models Methods Appl. Sci. 18, 1787–1827 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Harbrecht, H., Hohage, T.: Fast methods for three-dimensional inverse obstacle scattering problems. J Integr. Equ. Appl, 237–260 (2007)

  46. Hettlich, F.: Fréchet derivatives in inverse obstacle scattering. Inverse Probl. 11, 371 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. Hettlich, F.: Fréchet derivatives in inverse obstacle scattering. Inverse Probl. 14, 209–210 (1998)

    Article  MathSciNet  Google Scholar 

  48. Ho, K.L., Greengard, L.: A fast direct solver for structured linear systems by recursive skeletonization. SIAM J. Sci. Comput. 34, A2507–A2532 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Hu, G., Liu, X., Qu, F., Zhang, B.: Variational approach to scattering by unbounded rough surfaces with neumann and generalized impedance boundary conditions. Commun. Math. Sci. 13, 511–537 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ivanyshyn, O., Kress, R.: Inverse scattering for surface impedance from phase-less far field data. J. Comput. Phys. 230, 3443–3452 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kirsch, A.: Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Probl. 14, 1489 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kress, R.: Integral equation methods in inverse obstacle scattering with a generalized impedance boundary condition. In: Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan, pp 721–740. Springer (2018)

  53. Some old and some new results in inverse obstacle scattering, Maxwell’s Equations: Analysis and Numerics 24 (2019)

  54. Kress, R., Rundell, W.: Inverse scattering for shape and impedance. Inverse Probl. 17, 1075 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  55. Kress, R., Rundell, W.: Inverse scattering for shape and impedance revisited. J. Integr. Equ. Appl. 30, 293–311 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  56. Kuchment, P.: The Radon Transform and Medical Imaging, CBMS-NSF Regional Conference Series in Applied Mathematics Society for Industrial and Applied Mathematics (2014)

  57. Lechleiter, A., Haddar, H.: Generalized impedance boundary conditions for rough surface scattering. In: Proceedings of waves, vol. 2009 (2009)

  58. Lee, K.M.: An inverse scattering problem from an impedance obstacle. J. Comput. Phys. 227, 431–439 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. Martinsson, P.G., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205, 1–23 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. Nguyen, H.M., Nguyen, L.V.: Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation. Math. Models Methods Appl. Sci. 25, 1927–1960 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  61. Qin, H.H., Colton, D.: The inverse scattering problem for cavities with impedance boundary condition. Adv. Comput. Math. 36, 157–174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  62. Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Probl. 22, 663 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  63. Sini, M., Thanh, N.T., Rundell, W.: Inverse acoustic obstacle scattering using multifrequency measurements. Inverse Probl. Imaging, 6 (2012)

  64. Smith, R.T.: An inverse acoustic scattering problem for an obstacle with an impedance boundary condition. J. Math. Anal. Appl. 105, 333–356 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  65. Ustinov, E.: Encyclopedia of Remote Sensing, pp 247–251. Springer, New York, New York, NY (2014). ch. Geophysical Retrieval, Inverse Problems in Remote Sensing

    Book  Google Scholar 

  66. Yaman, O.I.: Reconstruction of generalized impedance functions for 3d acoustic scattering. J. Comput. Phys. 392, 444–455 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  67. Yang, J., Zhang, B., Zhang, H.: Reconstruction of complex obstacles with generalized impedance boundary conditions from far-field data. SIAM J. Appl. Math. 74, 106–124 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Leslie Greengard for many useful discussions.

Funding

C. Borges was supported in part by the Office of Naval Research under award number N00014-21-1-2389.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Carlos Borges

  2. Center for Computational Mathematics, Flatiron Institute, New York, NY, 10010, USA

    Manas Rachh

Authors
  1. Carlos Borges
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manas Rachh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manas Rachh.

Ethics declarations

Competing interests

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

Additional information

Communicated by: Michael O’Neil

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection: Advances in Computational Integral Equations Guest Editors: Stephanie Chaillat, Adrianna Gillman, Per-Gunnar Martinsson, Michael O’Neil, Mary-Catherine Kropinski, Timo Betcke, Alex Barnett

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borges, C., Rachh, M. Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization. Adv Comput Math 48, 2 (2022). https://doi.org/10.1007/s10444-021-09915-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09915-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation