Numerical resolution of an electromagnetic inverse medium problem at fixed frequency

The aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency. The governing equations are the time-harmonic Maxwell equations formulated in electric field in a two-dimensional bounded domain. We express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields. Our numerical reconstruction algorithm combines the BFGS method and an iterative process, called the Adaptive Eigenspace Inversion. The unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator. Both the eigenspace and the mesh are iteratively adapted during the minimization procedure. Numerical experiments illustrate the performance of the reconstruction for various configurations.

[1]  A. Calderón,et al.  On an inverse boundary value problem , 2006 .

[2]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[3]  Peter Monk,et al.  Recent Developments in Inverse Acoustic Scattering Theory , 2000, SIAM Rev..

[4]  H. Ammari,et al.  Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations , 2001 .

[5]  Thorsten Hohage,et al.  Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem , 2006, J. Comput. Phys..

[6]  L. Beilina,et al.  An adaptive finite element method in reconstruction of coefficients in Maxwell’s equations from limited observations , 2015, Applications of Mathematics.

[7]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[8]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[9]  Sergey Kabanikhin,et al.  Inverse Problems for Maxwell's Equations , 1994 .

[10]  Francesca Rapetti,et al.  Numerical Modeling and High-Speed Parallel Computing: New Perspectives on Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring. , 2017, IEEE Antennas and Propagation Magazine.

[11]  Gang Bao,et al.  Numerical solution of an inverse medium scattering problem for Maxwell's Equations at fixed frequency , 2009, J. Comput. Phys..

[12]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[13]  Marion,et al.  NUMERICAL RECONSTRUCTION OF SMALL PERTURBATIONS IN THE ELECTROMAGNETIC COEFFICIENTS OF A DIELECTRIC MATERIAL , 2014 .

[14]  Peter Monk,et al.  The linear sampling method for solving the electromagnetic inverse medium problem , 2002 .

[15]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[16]  Mordecai Avriel,et al.  Nonlinear programming , 1976 .

[17]  Gunther Uhlmann,et al.  Inverse problems for the anisotropic Maxwell equations , 2009, 0905.3275.

[18]  Petri Ola,et al.  Inverse Boundary Value Problem for Maxwell Equations with Local Data , 2009 .

[19]  Denis S. Grebenkov,et al.  Localization of Laplacian Eigenfunctions in Circular, Spherical, and Elliptical Domains , 2012, SIAM J. Appl. Math..

[20]  Maya de Buhan,et al.  Logarithmic stability in determination of a 3D viscoelastic coefficient and a numerical example , 2010 .

[21]  C Gabriel,et al.  The dielectric properties of biological tissues: I. Literature survey. , 1996, Physics in medicine and biology.

[22]  M. Vogelius,et al.  Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogen , 2000 .

[23]  R. W. Lau,et al.  The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. , 1996, Physics in medicine and biology.

[24]  L. B. Slichter.  An Inverse Boundary Value Problem in Electrodynamics , 1933 .

[25]  P. George,et al.  Mesh Generation: Application to Finite Elements , 2007 .

[26]  Ekaterina Iakovleva,et al.  MUSIC-Type Electromagnetic Imaging of a Collection of Small Three-Dimensional Inclusions , 2007, SIAM J. Sci. Comput..

[27]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[28]  Maya de Buhan,et al.  A new approach to solve the inverse scattering problem for waves: combining the TRAC and the adaptive inversion methods , 2013 .

[29]  L. Beilina.  Adaptive finite element method for a coefficient inverse problem for Maxwell's system , 2011 .