Inverse homogenization: Can one hear the structure of a composite?

Abstract: Inverse homogenization is a problem of deriving information about the microgeometry of a finely structured medium from its known effective properties. I will discuss an approach to this problem based on reconstructing the matrix-valued spectral measure in the Stieltjes integral representation of the effective properties of a two-component composite. This integral representation relates the n-point correlation functions of the microstructure to the moments of the spectral measure of an operator depending on the composite’s geometry. I will show that the spectral measure which contains all information about the microstructure, can be uniquely recovered from frequency dependent effective data; this allows to view the problem as an inverse spectral problem. In particular, the moments of the measure and the spectral gaps at the ends of the spectral interval can be uniquely reconstructed, which results in the unique identification of the volume fractions of materials in the composite and estimates for the connectedness of its phases. I will discuss the recovery of microstructural parameters from electromagnetic and viscoelastic effective measurements and show that the resulting spectroscopic imaging method provides an efficient way to construct spectrally matched microstructures.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine