Dehomogenization: reconstruction of moments of the spectral measure of the composite

This paper deals with the inverse homogenization or dehomogenization problem of recovering geometric information about the structure of a two-component composite medium from the effective complex permittivity of the composite. The approach is based on the reconstruction of moments of the spectral measure in the Stieltjes analytic representation of the effective property. The moments of the spectral measure are linked to n-point correlation functions of the structure of the composite and thus contain information about the microgeometry. We show that the moments can be uniquely recovered from the measurements of the effective property in a range of frequencies. Two methods of numerical reconstruction of the moments are developed and analyzed. One method, which is referred to as a direct method of moment reconstruction, is based on the solution of the Vandermonde system arising in series expansion of the Stieltjes integral. The second, indirect, method reformulates the problem and reduces it to the problem of reconstruction of the spectral function. This last problem is ill-posed and requires regularization. We show that even though the reconstructed spectral function can be quite sensitive to the choice of the regularization scheme, the moments of the spectral functions can be stably reconstructed. The applicability of these two methods in terms of the choice of data points is also discussed in this paper.