Recovery of multiple obstacles by probe method

Cheng, Jin; Liu, Jijun; Nakamura, Gen; Wang, Shengzhang

doi:10.1090/S0033-569X-09-01101-0

Authors: Jin Cheng, Jijun Liu, Gen Nakamura and Shengzhang Wang
Journal: Quart. Appl. Math. 67 (2009), 221-247
MSC (2000): Primary 35R30, 35J05, 76Q05
DOI: https://doi.org/10.1090/S0033-569X-09-01101-0
Published electronically: March 27, 2009
MathSciNet review: 2514633
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider an inverse scattering problem for multiple obstacles $D=\bigcup _{j=1}^ND_j\subset {R}^3$ with different types of boundary for $D_j$. By constructing an indicator function from the far-field pattern of the scattered wave, we can firstly reconstruct the shape of all obstacles, then identify the type of boundary for each obstacle, as well as the boundary impedance in the case that obstacles have the Robin-type boundary condition. The novelty of our probe method compared with the existing probe method is that we succeeded in identifying the type of boundary condition for multiple obstacles by analyzing the behavior of both the imaginary part and the real part of the indicator function. The numerical realizations are given to show the performance of this inversion method.

References

Jin Cheng, Jijun Liu, and Gen Nakamura, Recovery of the shape of an obstacle and the boundary impedance from the far-field pattern, J. Math. Kyoto Univ. 43 (2003), no. 1, 165–186. MR 2028705, DOI https://doi.org/10.1215/kjm/1250283745
J. Cheng, J. J. Liu, and G. Nakamura, The numerical realization of the probe method for the inverse scattering problems from the near-field data, Inverse Problems 21 (2005), no. 3, 839–855. MR 2144808, DOI https://doi.org/10.1088/0266-5611/21/3/004
David Colton and Andreas Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems 12 (1996), no. 4, 383–393. MR 1402098, DOI https://doi.org/10.1088/0266-5611/12/4/003
David Colton and Andreas Kirsch, The determination of the surface impedance of an obstacle from measurements of the far field pattern, SIAM J. Appl. Math. 41 (1981), no. 1, 8–15. MR 622870, DOI https://doi.org/10.1137/0141002
David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980
Michael A. Epton and Benjamin Dembart, Multipole translation theory for the three-dimensional Laplace and Helmholtz equations, SIAM J. Sci. Comput. 16 (1995), no. 4, 865–897. MR 1335895, DOI https://doi.org/10.1137/0916051
Klaus Erhard and Roland Potthast, A numerical study of the probe method, SIAM J. Sci. Comput. 28 (2006), no. 5, 1597–1612. MR 2272180, DOI https://doi.org/10.1137/040607149
N. I. Grinberg, Obstacle visualization via the factorization method for the mixed boundary value problem, Inverse Problems 18 (2002), no. 6, 1687–1704. MR 1955913, DOI https://doi.org/10.1088/0266-5611/18/6/317
Masaru Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems 14 (1998), no. 4, 949–954. MR 1642552, DOI https://doi.org/10.1088/0266-5611/14/4/012
Masaru Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations 23 (1998), no. 7-8, 1459–1474. MR 1642619, DOI https://doi.org/10.1080/03605309808821390
M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion 30 (1999), no. 3, 205–223. MR 1708203, DOI https://doi.org/10.1016/S0165-2125%2899%2900006-2
Masaru Ikehata and Gen Nakamura, Slicing of a three-dimensional object from boundary measurements, Inverse Problems 15 (1999), no. 5, 1243–1253. MR 1715362, DOI https://doi.org/10.1088/0266-5611/15/5/309
Masaru Ikehata, Gen Nakamura, and Kazumi Tanuma, Identification of the shape of the inclusion in the anisotropic elastic body, Appl. Anal. 72 (1999), no. 1-2, 17–26. MR 1775432, DOI https://doi.org/10.1080/00036819908840727
Andreas Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems 14 (1998), no. 6, 1489–1512. MR 1662460, DOI https://doi.org/10.1088/0266-5611/14/6/009
Andrzej W. Kędzierawski, The determination of the surface impedance of an obstacle, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 1–15. MR 1200183, DOI https://doi.org/10.1017/S001309150000585X
N. I. Grinberg and A. Kirsch, The factorization method for obstacles with a priori separated sound-soft and sound-hard parts, Math. Comput. Simulation 66 (2004), no. 4-5, 267–279. MR 2079578, DOI https://doi.org/10.1016/j.matcom.2004.02.011
Rainer Kress and Axel Zinn, On the numerical solution of the three-dimensional inverse obstacle scattering problem, J. Comput. Appl. Math. 42 (1992), no. 1, 49–61. MR 1181580, DOI https://doi.org/10.1016/0377-0427%2892%2990162-Q
Rainer Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math. 61 (1995), no. 3, 345–360. MR 1361270, DOI https://doi.org/10.1016/0377-0427%2894%2900073-7
Jijun Liu, Inverse scattering for the problem with impedance-type boundary, J. Partial Differential Equations 13 (2000), no. 3, 279–288. MR 1779075
Jijun Liu, Determination of Dirichlet-to-Neumann map for a mixed boundary problem, Appl. Math. Comput. 161 (2005), no. 3, 843–864. MR 2113524, DOI https://doi.org/10.1016/j.amc.2003.12.043
Lars Mönch, A Newton method for solving the inverse scattering problem for a sound-hard obstacle, Inverse Problems 12 (1996), no. 3, 309–323. MR 1391541, DOI https://doi.org/10.1088/0266-5611/12/3/010
Roland Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources, J. Comput. Appl. Math. 114 (2000), no. 2, 247–274. MR 1737077, DOI https://doi.org/10.1016/S0377-0427%2899%2900201-0
Robert T. Smith, An inverse acoustic scattering problem for an obstacle with an impedance boundary condition, J. Math. Anal. Appl. 105 (1985), no. 2, 333–356. MR 778470, DOI https://doi.org/10.1016/0022-247X%2885%2990052-6