Inverse electromagnetic problem for microstructured media

The focus of this dissertation is on studies of the inverse electromagnetic problem for media with a microstructure. It deals with reconstruction of information about the structure of a two-component composite material from its known effective complex permittivity. It is assumed that the heterogeneous material occupying the set is E-periodic and the scale of microstructure is much smaller than the wavelength of the electromagnetic signal. A detailed analysis of the analytic Stieltjes representation of the effective permittivity tensor of the composite is included. The spectral function contains information about the microgeometry of the composite in Stieltjes integral representation of the effective permittivity, and the spectral measure can be uniquely determined if the spectral function is known on an arc in the complex plane. However, from the computation point of view, the problem of reconstruction of the spectral measure is extremely ill-posed. In order to develop a stable numerical algorithm for the approximation of the spectral function, a new inversion technique based on the constrained rational (Pade) approximation of spectral functions is developed for the reconstruction of the spectral measure in the analytic Stieltjes representation of the effective permittivity tensor. The rational approximation of the spectral function is obtained from the solution of a constrained minimization problem followed by its partial fractions decomposition. To demonstrate the validity of the algorithm, the inversion method is applied to the reconstruction of spectral functions of composites with different structures corresponding to known analytical models. The results are used in the problem of estimation of the volume fractions of the constituents in a finely structured heterogeneous mixture, and in the problem of evaluation of moments of spectral function for equivalent classes of micro-geometries of composites. The method is also applied to the problem of spectral coupling of different effective properties. Numerical experiments for recovering the volume fraction of air in air-liquid mixtures and reconstruction of the moments of spectral functions for metal-insulator composite materials show good agreements between theoretical and predicted values. The approach can be used to evaluate material (structure) properties from measured complex permittivity or other properties of the composite.