Inverse homogenization for evaluation of effective properties of a mixture

The paper deals with indirect evaluation of the effective thermal or hydraulic conductivity of a random mixture of two different materials from the known effective complex permittivity of the same mixture. The method is based on deriving information about the microstructure of the composite from measurements of its effective properties; we call this approach inverse homogenization. This structural information is contained in the spectral measure in the Stieltjes representation of the effective complex permittivity. The spectral measure can be reconstructed from effective measurements and used to estimate other effective properties of the same material. We introduce S-equivalence of the geometric structures corresponding to the same spectral measure, and show that the microstructures of different mixtures can be distinguished by the homogenized measurements up to the introduced equivalence. We show that the identification problem for the spectral function has a unique solution, however, the problem is extremely ill-posed. Several stabilization techniques are discussed such as quadratically constrained minimization and reconstruction in the class of functions of bounded variation. The approach is applicable to porous media, biological materials, artificial composites and other heterogeneous materials in which the scale of microstructure is much smaller than the wavelength of the electromagnetic signal.