| Title:
|
Inverse scattering via nonlinear integral equations method for a sound-soft crack with phaseless data (English) |
| Author:
|
Gao, Peng |
| Author:
|
Dong, Heping |
| Author:
|
Ma, Fuming |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
149-165 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
inverse scattering problem |
| Keyword:
|
Helmholtz equation |
| Keyword:
|
crack |
| Keyword:
|
phaseless |
| Keyword:
|
translation invariance |
| MSC:
|
35J05 |
| MSC:
|
35P25 |
| MSC:
|
35R30 |
| MSC:
|
45E05 |
| MSC:
|
65R32 |
| MSC:
|
78A46 |
| idZBL:
|
Zbl 06890303 |
| idMR:
|
MR3795244 |
| DOI:
|
10.21136/AM.2018.0154-17 |
| . |
| Date available:
|
2018-05-09T08:53:47Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147187 |
| . |
| Reference:
|
[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. Zbl 1338.35490, MR 3505314, 10.1137/15M1043959 |
| Reference:
|
[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. Zbl 1351.65083, MR 3535661, 10.1088/0266-5611/32/8/085002 |
| Reference:
|
[3] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory.Applied Mathematical Sciences 93, Springer, New York (2013). Zbl 1266.35121, MR 2986407, 10.1007/978-1-4614-4942-3 |
| Reference:
|
[4] Ivanyshyn, O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern.Inverse Probl. Imaging 1 (2007), 609-622. Zbl 1194.35502, MR 2350217, 10.3934/ipi.2007.1.609 |
| Reference:
|
[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. Zbl 1135.65392, MR 2363789, 10.1216/jiea/1190905488 |
| Reference:
|
[6] Ivanyshyn, O., Kress, R.: Inverse scattering for planar cracks via nonlinear integral equations.Math. Methods Appl. Sci. 31 (2008), 1221-1232. Zbl 1153.65367, MR 2426204, 10.1002/mma.970 |
| Reference:
|
[7] Ivanyshyn, O., Kress, R.: Identification of sound-soft 3D obstacles from phaseless data.Inverse Probl. Imaging 4 (2010), 131-149. Zbl 1220.35194, MR 2592786, 10.3934/ipi.2010.4.131 |
| Reference:
|
[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. Zbl 1121.76059, MR 2309563, 10.1093/imamat/hxl026 |
| Reference:
|
[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. Zbl 1255.65203, MR 2980733, 10.1016/j.apnum.2012.05.011 |
| Reference:
|
[10] Kirsch, A., Ritter, S.: A linear sampling method for inverse scattering from an open arc.Inverse Probl. 16 (2000), 89-105. Zbl 0968.35129, MR 1741229, 10.1088/0266-5611/16/1/308 |
| Reference:
|
[11] Kress, R.: Fréchet differentiability of the far field operator for scattering from a crack.J. Inverse Ill-Posed Probl. 3 (1995), 305-313. Zbl 0846.35146, MR 1366555, 10.1515/jiip.1995.3.4.305 |
| Reference:
|
[12] Kress, R.: Inverse scattering from an open arc.Math. Methods Appl. Sci. 18 (1995), 267-293. Zbl 0824.35030, MR 1319999, 10.1002/mma.1670180403 |
| Reference:
|
[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. Zbl 0880.65105, MR 1603907, 10.1007/978-3-7091-6521-8_7 |
| Reference:
|
[14] Kress, R., Serranho, P.: A hybrid method for two-dimensional crack reconstruction.Inverse Probl. 21 (2005), 773-784. Zbl 1070.35126, MR 2146288, 10.1088/0266-5611/21/2/020 |
| Reference:
|
[15] Lee, K.-M.: Shape reconstructions from phaseless data.Eng. Anal. Bound. Elem. 71 (2016), 174-178. MR 3539835, 10.1016/j.enganabound.2016.08.001 |
| Reference:
|
[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. Zbl 1185.35329, MR 2587064, 10.1016/j.jmaa.2009.11.031 |
| Reference:
|
[17] Mönch, L.: On the inverse acoustic scattering problem by an open arc: the sound-hard case.Inverse Probl. 13 (1997), 1379-1392. Zbl 0894.35079, MR 1474374, 10.1088/0266-5611/13/5/017 |
| Reference:
|
[18] Yan, Y., Sloan, I. H.: On integral equations of the first kind with logarithmic kernels.J. Integral Equations Appl. 1 (1988), 549-579. Zbl 0682.45001, MR 1008406, 10.1216/JIE-1988-1-4-549 |
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