Article
MSC:
35J05,
35P25,
35R30,
45E05,
65R32,
78A46 | MR 3795244 | Zbl 06890303 | DOI: 10.21136/AM.2018.0154-17
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
inverse scattering problem; Helmholtz equation; crack; phaseless; translation invariance
inverse scattering problem; Helmholtz equation; crack; phaseless; translation invariance
Summary:
We consider the inverse scattering of time-harmonic plane waves to reconstruct the shape of a sound-soft crack from a knowledge of the given incident field and the phaseless data, and we check the invariance of far field data with respect to translation of the crack. We present a numerical method that is based on a system of nonlinear and ill-posed integral equations, and our scheme is easy and simple to implement. The numerical implementation is described and numerical examples are presented to illustrate the feasibility of the proposed method.
References:
[1] Ammari, H., Chow, Y. T., Zou, J.: Phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients. SIAM J. Appl. Math. 76 (2016), 1000-1030. DOI 10.1137/15M1043959 | MR 3505314 | Zbl 1338.35490
[2] Bao, G., Zhang, L.: Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data. Inverse Probl. 32 (2016), Article ID 085002, 16 pages. DOI 10.1088/0266-5611/32/8/085002 | MR 3535661 | Zbl 1351.65083
[3] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences 93, Springer, New York (2013). DOI 10.1007/978-1-4614-4942-3 | MR 2986407 | Zbl 1266.35121
[4] Ivanyshyn, O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Probl. Imaging 1 (2007), 609-622. DOI 10.3934/ipi.2007.1.609 | MR 2350217 | Zbl 1194.35502
[5] Ivanyshyn, O., Johansson, T.: Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle. J. Integral Equations Appl. 19 (2007), 289-308. DOI 10.1216/jiea/1190905488 | MR 2363789 | Zbl 1135.65392
[6] Ivanyshyn, O., Kress, R.: Inverse scattering for planar cracks via nonlinear integral equations. Math. Methods Appl. Sci. 31 (2008), 1221-1232. DOI 10.1002/mma.970 | MR 2426204 | Zbl 1153.65367
[7] Ivanyshyn, O., Kress, R.: Identification of sound-soft 3D obstacles from phaseless data. Inverse Probl. Imaging 4 (2010), 131-149. DOI 10.3934/ipi.2010.4.131 | MR 2592786 | Zbl 1220.35194
[8] 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 (2007), 96-112. DOI 10.1093/imamat/hxl026 | MR 2309563 | Zbl 1121.76059
[9] Karageorghis, A., Johansson, B. T., Lesnic, D.: The method of fundamental solutions for the identification of a sound-soft obstacle in inverse acoustic scattering. Appl. Numer. Math. 62 (2012), 1767-1780. DOI 10.1016/j.apnum.2012.05.011 | MR 2980733 | Zbl 1255.65203
[10] Kirsch, A., Ritter, S.: A linear sampling method for inverse scattering from an open arc. Inverse Probl. 16 (2000), 89-105. DOI 10.1088/0266-5611/16/1/308 | MR 1741229 | Zbl 0968.35129
[11] Kress, R.: Fréchet differentiability of the far field operator for scattering from a crack. J. Inverse Ill-Posed Probl. 3 (1995), 305-313. DOI 10.1515/jiip.1995.3.4.305 | MR 1366555 | Zbl 0846.35146
[12] Kress, R.: Inverse scattering from an open arc. Math. Methods Appl. Sci. 18 (1995), 267-293. DOI 10.1002/mma.1670180403 | MR 1319999 | Zbl 0824.35030
[13] Kress, R., Rundell, W.: Inverse obstacle scattering with modulus of the far field pattern as data. Inverse Problems in Medical Imaging and Nondestructive Testing H. W. Engl et al. Springer, Wien (1997), 75-92. DOI 10.1007/978-3-7091-6521-8_7 | MR 1603907 | Zbl 0880.65105
[14] Kress, R., Serranho, P.: A hybrid method for two-dimensional crack reconstruction. Inverse Probl. 21 (2005), 773-784. DOI 10.1088/0266-5611/21/2/020 | MR 2146288 | Zbl 1070.35126
[15] Lee, K.-M.: Shape reconstructions from phaseless data. Eng. Anal. Bound. Elem. 71 (2016), 174-178. DOI 10.1016/j.enganabound.2016.08.001 | MR 3539835
[16] Liu, X., Zhang, B.: Unique determination of a sound-soft ball by the modulus of a single far field datum. J. Math. Anal. Appl. 365 (2010), 619-624. DOI 10.1016/j.jmaa.2009.11.031 | MR 2587064 | Zbl 1185.35329
[17] Mönch, L.: On the inverse acoustic scattering problem by an open arc: the sound-hard case. Inverse Probl. 13 (1997), 1379-1392. DOI 10.1088/0266-5611/13/5/017 | MR 1474374 | Zbl 0894.35079
[18] Yan, Y., Sloan, I. H.: On integral equations of the first kind with logarithmic kernels. J. Integral Equations Appl. 1 (1988), 549-579. DOI 10.1216/JIE-1988-1-4-549 | MR 1008406 | Zbl 0682.45001
Search
Partner of
