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Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields

Carpio Rodríguez, Ana María and Johansson, B.T. and Rapún , M.L. (2010) Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields. Journal of Mathematical Imaging and Vision, 36 (2). pp. 185-199. ISSN 0924-9907 (Print) 1573-7683 (Online)

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Abstract

An inverse problem is considered where the structure of multiple sound-soft planar obstacles is to be determined given the direction of the incoming acoustic field and knowledge of the corresponding total field on a curve located outside the obstacles. A local uniqueness result is given for this inverse problem suggesting that the reconstruction can be achieved by a single incident wave. A numerical procedure based on the concept of the topological derivative of an associated cost functional is used to produce images of the obstacles. No a priori assumption about the number of obstacles present is needed. Numerical results are included showing that accurate reconstructions can be obtained and that the proposed method is capable of finding both the shapes and the number of obstacles with one or a few incident waves.

Item Type:Article
Uncontrolled Keywords:Inverse scattering; Reconstruction; Uniqueness; Shape reconstruction; Topological derivative; Scattering; Sound-soft obstacles; Acoustic waves
Subjects:Sciences > Physics > Mathematical physics
ID Code:14884
References:

1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions.

Dover, New York (1972)

2. Ben Hassen, F., Liu, J., Potthast, R.: On source analysis by wave

splitting with applications in inverse scattering of multiple obstacles.

J. Comput. Math. 25(3), 266–281 (2007)

3. Carpio, A., Rapun, M.L.: Topological Derivatives for Shape Reconstruction.

Lect. Not. Mat., vol. 1943, pp. 85–131 (2008)

4. Carpio, A., Rapun, M.L.: Solving inverse inhomogeneous problems

by topological derivative methods. Inverse Probl. 24, 045014

(2008)

5. Carpio, A., Rapun, M.L.: Domain reconstruction by thermal measurements.

J. Comput. Phys. 227(17), 8083–8106 (2008)

6. Carpio, A., Rapun, M.L.: An iterative method for parameter identification

and shape reconstruction. Inverse Probl. Sci. Eng. 18,

35–50 (2010)

7. Colton, D., Gieberman, K., Monk, P.: A regularized sampling

method for solving three dimensional inverse scattering problems.

SIAM J. Sci. Comput. 21, 2316–2330 (2000)

8. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering.

Springer, Berlin (1998)

9. Colton, D., Kress, R.: Integral Equation Methods in Scattering

Theory. Wiley, New York (1983)

10. Colton, D., Sleeman, B.D.: Uniqueness theorems for the inverse

problem of acoustic scattering. IMA J. Appl. Math. 31, 253–259

(1983)

11. Feijoo, G.R.: A new method in inverse scattering based on the

topological derivative. Inverse Probl. 20, 1819–1840 (2004)

12. Feijoo, G.R., Oberai, A.A., Pinsky, P.M.: An application of shape

optimization in the solution of inverse acoustic scattering problems.

Inverse Probl. 20, 199–228 (2004)

13. Gintides, D.: Local uniqueness for the inverse scattering problem

in acoustics via the Faber-Krahn inequality. Inverse Probl. 21,

1195–1205 (2005)

14. Guzina, B.B., Bonnet, M.: Small-inclusion asymptotic of misfit

functionals for inverse problems in acoustics. Inverse Probl. 22,

1761–1785 (2006)

15. Guzina, B.B., Chikichev, I.: From imaging to material identification:

A generalized concept of topological sensitivity. J. Mech.

Phys. Solids 55, 245–279 (2007)

16. Ivanyshyn, O., Johansson, T.: Nonlinear integral equations methods

for the reconstruction of an acoustically sound-soft obstacle.

J. Integral Equ. Appl. 19(3), 289–308 (2007)

17. Johansson, T., Sleeman, B.D.: Reconstruction of an acoustically

sound-soft obstacle from one incident field and the far field pattern.

IMA J. Appl. Math. 72, 96–112 (2007)

18. Keller, J.B., Givoli, D.: Exact non-reflecting boundary conditions.

J. Comput. Phys. 82, 172–192 (1989)

19. Kirsch, A.: The domain derivative and two applications in inverse

scattering theory. Inverse Probl. 9, 81–96 (1993)

20. Kress, R., Rundell, W.: Nonlinear integral equations and the iterative

solution for an inverse boundary value problem. Inverse Probl.

21, 1207–1223 (2005)

21. Linton, C.M., Martin, P.A.: Multiple scattering by random configuration

of circular cylinders: Second-order corrections for the effective

wavenumber. J. Acoust. Soc. Am. 117, 3413–3423 (2005)

22. Litman, A., Lesselier, D., Santosa, F.: Reconstruction of a two dimensional

binary obstacle by controlled evolution of a level set.

Inverse Probl. 14, 685–706 (1998)

23. Martin, P.A.: Multiple Scattering, Interaction of Time-Harmonic

Waves with N Obstacles. Cambridge Univ. Press, Cambridge

(2006)

24. Potthast, R.: A survey on sampling and probe methods for inverse

problems, Topical Review. Inverse Probl. 22, R1–R47 (2006)

25. Samet, B., Amstutz, S., Masmoudi, M.: The topological asymptotic

for the Helmholtz equation. SIAM J. Control Optim. 42,

1523–1544 (2003)

26. Santosa, F.: A level set approach for inverse problems involving

obstacles. ESAIM Control Optim. Calc. Var. 1, 17–33 (1996)

27. Sleeman, B.D.: The inverse problem of acoustic scattering. Applied

Mathematics Institute Technical Report No. 114 A, University

of Delaware, Newark, 1981

28. Twersky, V.: Multiple scattering of radiation by an arbitrary configuration

of parallel cylinders. J. Acoust. Soc. Am. 24, 42–46

(1952)

29. Twersky, V.: Multiple scattering of radiation by an arbitrary planar

configuration of parallel cylinders and by two parallel cylinders. J.

Appl. Phys. 23, 407–414 (1952)

30. Young, J.W., Bertrand, J.C.: Multiple scattering by two cylinders.

J. Acoust. Soc. Am. 58, 1190–1195 (1975)

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