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Recovering an electromagnetic obstacle by a few phaseless backscattering measurements. (English) Zbl 1432.78008

Summary: We consider the electromagnetic scattering from a convex polyhedral PEC or PMC obstacle due to a time-harmonic incident plane wave. It is shown that the modulus of the far-field pattern in the backscattering aperture possesses a certain local maximum behavior. Using the local maximum indicating phenomena, one can determine the exterior unit normal directions, as well as the face areas, of the front faces of the obstacle. Then we propose a recovery scheme of reconstructing the obstacle by phaseless backscattering measurements. This work significantly extends our recent study in [the first two authors, J. Differ. Equations 259, No. 5, 2101–2120 (2015; Zbl 1330.78013)] from two dimensions and acoustic scattering to the more challenging three dimensions and electromagnetic scattering.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
35Q61 Maxwell equations
78M50 Optimization problems in optics and electromagnetic theory

Citations:

Zbl 1330.78013

Software:

DistMesh; Qhull
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Full Text: DOI arXiv

References:

[1] Ammari H and Kang H 2004 Reconstruction of Small Inhomogeneities from Boundary Measurements(Lecture Notes in Mathematics vol 1846) (Berlin: Springer) · Zbl 1113.35148
[2] Ammari H and Kang H 2007 Polarization, Moment Tensors (New York: Springer) · Zbl 1220.35001
[3] BBarber C, Dobkin D P and Huhdanpaa H 1996 The quickhull algorithm for convex hulls ACM Trans. Math. Softw.22 469–83 · Zbl 0884.65145
[4] Chandler-Wilde S N and Langdon S 2014 Acoustic scattering: high frequency boundary element methods and unified transform methods Unified Transform for Boundary Value Problems (Philadelphia, PA: SIAM) pp 181–226 · Zbl 1346.65065
[5] Colton D and Kress R 1998 Inverse Acoustic, Electromagnetic Scattering Theory 2nd edn (Berlin: Springer)
[6] Hewett D P, Langdon S and Melenk J M 2013 A high frequency hp boundary element method for scattering by convex polygons SIAM J. Numer. Anal.51 629–53 · Zbl 1267.65191
[7] Isakov V 2006 Inverse Problems for Partial Differential Equations(Applied Mathematical Sciences vol 127) 2nd edn (New York: Springer)
[8] Klain D A 2004 The Minkowski problem for polytopes Adv. Math.185 270–88 · Zbl 1053.52015
[9] Kunis S and Potts D 2003 Fast spherical Fourier algorithms J. Comput. Appl. Math.161 75–88 · Zbl 1033.65123
[10] Kress R and Rundell W 1997 Inverse obstacle scattering with modulus of the far field pattern as data Inverse Problems in Medical Imaging and Nondestructive Testing(Oberwolfach, 1996) (Vienna: Springer) pp 75–92 · Zbl 0880.65105
[11] Lax P D and Phillips R S 1967 Scattering Theory (New York: Academic)
[12] Li J and Liu H 2014 Recovering a polyhedral obstacle by a few backscattering measurements J. Differential Equations259 2101–20
[13] Li J, Liu H, Shang Z and Sun H 2013 Two single-shot methods for locating multiple electromagnetic scatterers SIAM J. Appl. Math.73 1721–46 · Zbl 1323.78012
[14] Li J, Liu H and Wang Q 2013 Locating multiple multi-scale electromagnetic scatterers by a single far-field measurement SIAM J. Imaging Sci.6 2285–309 · Zbl 1302.78016
[15] Liu H 2008 A global uniqueness for formally determined inverse electromagnetic obstacle scattering Inverse Problems24 035018
[16] Liu H, Yamamoto M and Zou J 2007 Reflection principle for Maxwell’s equations, its application to inverse electromagnetic scattering problem Inverse Problems23 2357–66 · Zbl 1126.35073
[17] Majda A 1976 High frequency asymptotics for the scattering matrix, the inverse problem of acoustical scattering Commun. Pure Appl. Math.29 261–91 · Zbl 0463.35048
[18] McLean W 2000 Strongly Elliptic Systems, Boundary Integral Equations (Cambridge: Cambridge University Press)
[19] Melrose R B aand Taylor M E 1985 Near peak scattering, the corrected Kirchhoff approximation for a convex obstacle Adv. Math.55 242–315 · Zbl 0591.58034
[20] Nedelec J C 2001 Acoustic, Electromagnetic Equations (New York: Springer)
[21] Persson P-O and Strang G 2004 A simple mesh generator in MATLAB SIAM Rev.46 329–45 · Zbl 1061.65134
[22] Uhlmann G 2003 Inside Out: Inverse Problems, Applications (Cambridge: Cambridge University Press)
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