Solution of electromagnetic inverse medium scattering problems by the adaptive finite element method and perfectly matched layer

Abstract

The paper presents a study of solution of inverse medium scattering problems for time-harmonic electromagnetics in 3D. The solution is based on minimization of the misfit function between the observed scattered waves and trial solutions of direct problems for the sought approximate distribution of electric permittivity. The sources of illuminating waves and the observation points of the scattered fields are located in the vicinity of the scatterer. The simulations of the direct problems are performed with the adaptive finite element method. Tests with real- and complex-valued distributions of electric permittivity, and with complete and incomplete measurement data are presented.

Keywords:

• inverse problems
• electromagnetic field
• finite element methods
• optimization

Classification:

• 65N30
• 78A46

Acknowledgments

The work of W. Rachowicz was supported by Polish Ministry of Higher Education, under grant N N519 405 234.

Notes

Notes

1. In fact the introduction of tensors and results from application of an auxiliary extension of Maxwell’s equations to complex arguments, called complex stretching which modifies both outgoing and incoming waves. The former decrease exponentialy while the latter grow exponentialy. One can use then the homogeneous Dirichlet boundary conditions to zero out the incoming waves without affecting the outgoing ones.

2. The form of the misfit function reflects the basic fact that the knowledge of the DtN-operator on a selected surface around the scatterer implies uniqueness of the solution of the inverse problem.[18] Of course we might have a chance to recover the DtN-operator if we tested at all points of the solutions corresponding to infinite number of sources located over some surface.