A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles


  • Frédérique Le Louër Université de Technologie de Compiègne




Maxwell equations, multiple scattering, inverse problems, fast solver, Gauss-Newton method


We consider fast and accurate solution methods for the direct and inverse scattering problems by a few three dimensional piecewise homogeneous dielectric obstacles around the resonance region. The forward problem is reduced to a system of second kind boundary integral equations. For the numerical solution of these coupled integral equations we modify a fast and accurate spectral algorithm, proposed by Ganesh and Hawkins [doi:10.1016/j.jcp.2008.01.016], by transporting these equations onto the unit sphere using the Piola transform of the boundary parametrisations. The computational performances of the forward solver are demonstrated on numerical examples for a variety of three-dimensional smooth and non smooth obstacles. The algorithm, that requires the knowledge of the boundary parametrisation and leads to invert small linear systems, is well-suited for the use of geometric optimisation tools to solve the inverse problem of recovering the shape of scatterers from the knowledge of noisy data. Computational details for the application of the iteratively regularised Gauss--Newton method to the numerical solution of the electromagnetic inverse problem are presented. Numerical experiments for the shape detection of multiple obstacles using incomplete radiation pattern data from back and front side are provided. The results in this article can also be applied for solving shape optimisation problems relying on time-harmonic electromagnetic waves. References
  • A. Altundag, On a Two-Dimensional Inverse Scattering Problem for a Dielectric, PhD thesis, University of G\T1\oe ttingen, 2012.
  • K. E. Atkinson, The numerical solution of Laplace's equation in three dimensions, SIAM J. Numer. Anal., 19 (1982), pp. 263–274. doi:10.1137/0719017
  • A. B. Bakushinski\T1\i , On a convergence problem of the iterative-regularized Gauss–Newton method, Comput. Math. Math. Phys, 32 (1992), pp. 1503–1509.
  • A. Buffa, R. Hiptmair, T. von Petersdorff, and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math., 95 (2003), pp. 459–485. doi:10.1007/s00211-002-0407-z
  • F. Cakoni, D. Colton, and P. Monk, The linear sampling method in inverse electromagnetic scattering, vol. 80 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2011.
  • A. Carpio, B. T. Johansson, and M.-L. Rapun, Determining planar multiple sound-soft obstacles from scattered acoustic fields, J. Math. Imaging Vision, 36 (2010), pp. 185–199. doi:10.1007/s10851-009-0182-x
  • D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 of Applied Mathematical Sciences, Springer, New York, third ed., 2013.
  • M. Costabel and F. Le Louer, On the Kleinman–Martin integral equation method for electromagnetic scattering by a dielectric body, SIAM J. Appl. Math., 71 (2011), pp. 635–656. doi:10.1137/090779462
  • M. Costabel and F. Le Louer, Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: Shape differentiability of pseudo-homogeneous boundary integral operators, Integr. Equ. Oper. Theory, 72 (2012), pp. 509–535. doi:10.1007/s00020-012-1954-z
  • M. Costabel and F. Le Louer, Shape derivatives of boundary integral operators in electromagnetic scattering. Part II: Application to scattering by a homogeneous dielectric obstacle, Integr. Equ. Oper. Theory, 73 (2012), pp. 17–48. doi:10.1007/s00020-012-1955-y
  • A. de La Bourdonnaye, Decomposition de \(H^-1/2_\protect \unhbox \voidb@x \hbox div(\Gamma )\) et nature de l'operateur de Steklov–Poincare du probleme exterieur de l'electromagnetisme, C. R. Acad. Sci. Paris Ser. I Math., 316 (1993), pp. 369–372.
  • M. Ganesh and I. G. Graham, A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys., 198 (2004), pp. 211–242. doi:10.1016/j.jcp.2004.01.007
  • M. Ganesh and S. C. Hawkins, A spectrally accurate algorithm for electromagnetic scattering in three dimensions, Numer. Algorithms, 43 (2006), pp. 25–60. doi:10.1007/s11075-006-9033-7
  • M. Ganesh and S. C. Hawkins, An efficient surface integral equation method for the time-harmonic Maxwell equations, ANZIAM J., 48 (2007), pp. C17–C33. doi:10.21914/anziamj.v48i0.60
  • M. Ganesh and S. C. Hawkins, A hybrid high-order algorithm for radar cross section computations, SIAM J. Sci. Comput., 29 (2007), pp. 1217–1243. doi:10.1137/060664859
  • M. Ganesh and S. C. Hawkins, A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces, J. Comput. Phys., 227 (2008), pp. 4543–4562. doi:10.1016/j.jcp.2008.01.016
  • M. Ganesh and S. C. Hawkins, A high-order algorithm for multiple electromagnetic scattering in three dimensions, Numer. Algorithms, 50 (2009), pp. 469–510. doi:10.1007/s11075-008-9238-z
  • M. Ganesh and S. C. Hawkins, An efficient \(O(N)\) algorithm for computing \(O(N^2)\) acoustic wave interactions in large \(N\)-obstacle three dimensional configurations, BIT, 55 (2015), pp. 117–139. doi:10.1007/s10543-014-0491-3
  • M. Ganesh, S. C. Hawkins and D. Volkov, An all-frequency weakly-singular surface integral equation for electromagnetism in dielectric media: Reformulation and well-posedness analysis, J. Math. Anal. Appl., 412 (2014), pp. 277–300. doi:10.1016/j.jmaa.2013.10.059
  • H. Haddar and R. Kress, On the Frechet derivative for obstacle scattering with an impedance boundary condition, SIAM J. Appl. Math., 65 (2004), pp. 194–208 (electronic). doi:10.1137/S0036139903435413
  • P. Hahner, A uniqueness theorem for a transmission problem in inverse electromagnetic scattering, Inverse Problems, 9 (1993), pp. 667–678. doi:10.1088/0266-5611/9/6/005
  • H. Harbrecht and T. Hohage, Fast methods for three-dimensional inverse obstacle scattering problems, J. Integral Equations Appl., 19 (2007), pp. 237–260. doi:10.1216/jiea/1190905486
  • R. F. Harrington, Boundary integral formulations for homogeneous materials bodies, J. Electromagnetics Waves and Applications, 3 (1989), pp. 1–15. doi:10.1163/156939389X00016
  • F. Hettlich, Frechet derivatives in inverse obstacle scattering, Inverse Problems, 11 (1995), pp. 371–382. doi:10.1088/0266-5611/11/2/007
  • F. Hettlich, Erratum: ``Frechet derivatives in inverse obstacle scattering'' [Inverse Problems 11 (1995), no. 2, 371–382; MR1324650 (95k:35217)], Inverse Problems, 14 (1998), pp. 209–210. doi:10.1088/0266-5611/14/1/017
  • F. Hettlich, The domain derivative of time-harmonic electromagnetic waves at interfaces, Math. Methods Appl. Sci., 35 (2012), pp. 1681–1689. doi:10.1002/mma.2548
  • T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), pp. 1279–1299. doi:10.1088/0266-5611/13/5/012
  • T. Hohage, Iterative Methods in Inverse Obstacle Scattering: Regularization Theory of Linear and Nonlinear Exponentially Ill-Posed Problems, PhD thesis, University of Linz, 1999.
  • T. Hohage and C. Schormann, A Newton-type method for a transmission problem in inverse scattering, Inverse Problems, 14 (1998), pp. 1207–1227. doi:10.1088/0266-5611/14/5/008
  • O. Ivanyshyn and R. Kress, Identification of sound-soft 3D obstacles from phaseless data, Inverse Probl. Imaging, 4 (2010), pp. 131–149. doi:10.3934/ipi.2010.4.131
  • O. Ivanyshyn Yaman and F. Le Louer, Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems, Inverse Problems, 32 (2016), pp. 095003, 24. doi:10.1088/0266-5611/32/9/095003
  • A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), pp. 81–96. doi:10.1088/0266-5611/9/1/005
  • R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math., 48 (1988), pp. 307–325. doi:10.1137/0148016
  • R. Kress, Electromagnetic waves scattering : Scattering by obstacles, Scattering, (2001), pp. 191–210. Pike, E. R. and Sabatier, P. C., eds., Academic Press, London.
  • R. Kress and L. Paivarinta, On the far-field in obstacle scattering, SIAM J. Appl. Math., 59 (1999), pp. 1413–1426 (electronic). doi:10.1137/S0036139997332257
  • F. Le Louer, Optimisation de formes d'antennes lentilles integrees aux ondes millimetrique, PhDthesis, Univ. Rennes 1, 2009. http://tel.archives-ouvertes.fr/tel-00421863/fr/.
  • F. Le Louer, A high order spectral algorithm for elastic obstacle scattering in three dimensions, J. Comput. Phys., 279 (2014), pp. 1–17. doi:10.1016/j.jcp.2014.08.047
  • F. Le Louer, Spectrally accurate numerical solution of hypersingular boundary integral equations for three-dimensional electromagnetic wave scattering problems, J. Comput. Phys., 275 (2014), pp. 662–666. doi:10.1016/j.jcp.2014.07.022
  • D. W. Mackowski, Analysis of radiative scattering for multiple sphere configurations, Proc. Roy. Soc. London Ser. A, 433 (1991), pp. 599–614. doi:10.1098/rspa.1991.0066
  • P. A. Martin and P. Ola, Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 185–208. doi:10.1017/S0308210500021296
  • J. R. Mautz, A stable integral equation for electromagnetic scattering from homogeneous dielectric bodies, IEEE Trans. Antennas and Propagation, 37 (1989), pp. 1070–1071. doi: 10.1109/8.34145
  • C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves, Berlin-Springer, 1969.
  • J.-C. Nedelec, Acoustic and electromagnetic equations, vol. 144 of Applied Mathematical Sciences, Springer-Verlag, New York, 2001. Integral representations for harmonic problems.
  • S. Onaka, Simple equations giving shapes of various convex polyhedra: The regular polydedra and polyhedra composed of crystallographically low-index planes, Philosophical Magazine Letters, 86 (2006), pp. 175–183. doi:10.1080/09500830600603050
  • R. Potthast, Frechet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems, 10 (1994), pp. 431–447. doi:10.1088/0266-5611/10/2/016
  • R. Potthast, Domain derivatives in electromagnetic scattering, Math. Methods Appl. Sci., 19 (1996), pp. 1157–1175.
  • R. Potthast, Frechet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain, J. Inverse Ill-Posed Probl., 4 (1996), pp. 67–84. doi:10.1515/jiip.1996.4.1.67
  • R. Song, X. Ye, and X. Chen, Reconstruction of scatterers with four different boundary conditions by T-matrix method, Inverse Probl. Sci. Eng., 23 (2015), pp. 601–616. doi:10.1080/17415977.2014.923418





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