Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization

Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical microstructures present in body tissues or geological features. Mathematical homogenization and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macroscales we observe and the underlying microscale models we assume. Among the various applications benefiting from homogenization, electrical impedance tomography (EIT) images the electrical conductivity of a body by measuring electrical potentials consequential to electric currents applied to the exterior of the body. EIT is routinely used in breast cancer detection and cardiopulmonary imaging, where current flow in fine-scale tissues underlies the resulting coarse-scale images.