The factorization method for cracks in inhomogeneous media

Jun Guo; Guozheng Yan; Jing Jin; Junhao Hu

Applications of Mathematics (2017)

  • Volume: 62, Issue: 5, page 509-533
  • ISSN: 0862-7940

Abstract

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We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.

How to cite

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Guo, Jun, et al. "The factorization method for cracks in inhomogeneous media." Applications of Mathematics 62.5 (2017): 509-533. <http://eudml.org/doc/294159>.

@article{Guo2017,
abstract = {We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.},
author = {Guo, Jun, Yan, Guozheng, Jin, Jing, Hu, Junhao},
journal = {Applications of Mathematics},
keywords = {inverse scattering; factorization method; crack; inhomogeneous media},
language = {eng},
number = {5},
pages = {509-533},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The factorization method for cracks in inhomogeneous media},
url = {http://eudml.org/doc/294159},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Guo, Jun
AU - Yan, Guozheng
AU - Jin, Jing
AU - Hu, Junhao
TI - The factorization method for cracks in inhomogeneous media
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 5
SP - 509
EP - 533
AB - We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.
LA - eng
KW - inverse scattering; factorization method; crack; inhomogeneous media
UR - http://eudml.org/doc/294159
ER -

References

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  1. Hassen, F. Ben, Boukari, Y., Haddar, H., 10.1080/17415977.2012.686997, Inverse Probl. Sci. Eng. 21 (2013), 210-234. (2013) Zbl1267.78036MR3022417DOI10.1080/17415977.2012.686997
  2. Bondarenko, O., Kirsch, A., Liu, X., 10.1088/0266-5611/29/4/045010, Inverse Probl. 29 (2013), Article ID 045010, 19 pages. (2013) Zbl1273.65167MR3042086DOI10.1088/0266-5611/29/4/045010
  3. Boukari, Y., Haddar, H., 10.3934/ipi.2013.7.1123, Inverse Probl. Imaging 7 (2013), 1123-1138. (2013) Zbl1302.35430MR3180673DOI10.3934/ipi.2013.7.1123
  4. Cakoni, F., Colton, D., 10.1088/0266-5611/19/2/303, Inverse Probl. 19 (2003), 279-295. (2003) Zbl1171.35487MR1991789DOI10.1088/0266-5611/19/2/303
  5. Cakoni, F., Colton, D., 10.1007/3-540-31230-7, Interaction of Mechanics and Mathematics, Springer, Berlin (2006). (2006) Zbl1099.78008MR2256477DOI10.1007/3-540-31230-7
  6. Cakoni, F., Fares, M'B., Haddar, H., 10.1088/0266-5611/22/3/007, Inverse Probl. 22 (2006), 845-867. (2006) Zbl1099.35167MR2235641DOI10.1088/0266-5611/22/3/007
  7. Cakoni, F., Harris, I., 10.1088/0266-5611/31/2/025002, Inverse Probl. 31 (2015), Article ID 025002, 22 pages. (2015) Zbl1327.78016MR3303167DOI10.1088/0266-5611/31/2/025002
  8. Colton, D., Haddar, H., 10.1088/0266-5611/21/1/023, Inverse Probl. 21 (2005), 383-398. (2005) Zbl1086.35129MR2146182DOI10.1088/0266-5611/21/1/023
  9. Grisel, Y., Mouysset, V., Mazet, P.-A., Raymond, J.-P., 10.1088/0266-5611/28/5/055003, Inverse Probl. 28 (2012), Article ID 055003, 19 pages. (2012) Zbl1239.78008MR2915291DOI10.1088/0266-5611/28/5/055003
  10. Guo, J., Hu, J., Yan, G., 10.1515/jiip-2014-0073, J. Inverse Ill-Posed Probl. 24 (2016), 527-541. (2016) Zbl06641005MR3552500DOI10.1515/jiip-2014-0073
  11. Guo, J., Wu, Q., Yan, G., 10.1093/imamat/hxu049, IMA J. Appl. Math. 80 (2015), 1199-1218. (2015) Zbl1330.35436MR3384435DOI10.1093/imamat/hxu049
  12. Kirsch, A., 10.1002/mma.1670110605, Math. Methods Appl. Sci. 11 (1989), 789-804. (1989) Zbl0692.35073MR1021401DOI10.1002/mma.1670110605
  13. Kirsch, A., Grinberg, N., 10.1093/acprof:oso/9780199213535.001.0001, Oxford Lecture Series in Mathematics and Its Applications 36, Oxford University Press, Oxford (2008). (2008) Zbl1222.35001MR2378253DOI10.1093/acprof:oso/9780199213535.001.0001
  14. Kirsch, A., Päivärinta, L., 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P, Math. Methods Appl. Sci. 21 (1998), 619-651. (1998) Zbl0915.35116MR1615992DOI10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P
  15. Kirsch, A., Ritter, S., 10.1088/0266-5611/16/1/308, Inverse Probl. 16 (2000), 89-105. (2000) Zbl0968.35129MR1741229DOI10.1088/0266-5611/16/1/308
  16. Lechleiter, A., 10.3934/ipi.2009.3.123, Inverse Probl. Imaging 3 (2009), 123-138. (2009) Zbl1184.78020MR2558306DOI10.3934/ipi.2009.3.123
  17. McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge (2000). (2000) Zbl0948.35001MR1742312
  18. Nachman, A. I., Päivärinta, L., Teirilä, A., 10.1016/j.jfa.2007.06.020, J. Funct. Anal. 252 (2007), 490-516. (2007) Zbl1131.78004MR2360925DOI10.1016/j.jfa.2007.06.020
  19. Potthast, R., 10.1201/9781420035483, Chapman & Hall/CRC Research Notes in Mathematics 427, Chapman & Hall/CRC, Boca Raton (2001). (2001) Zbl0985.78016MR1853728DOI10.1201/9781420035483
  20. Prössdorf, S., Saranen, J., 10.4171/ZAA/483, Z. Anal. Anwend. 13 (1994), 683-695. (1994) Zbl0822.65095MR1305602DOI10.4171/ZAA/483
  21. Zeev, N., Cakoni, F., 10.1137/070711542, SIAM J. Appl. Math. 69 (2009), 1024-1042. (2009) Zbl1173.35741MR2476589DOI10.1137/070711542

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