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 micro-structures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macro-scales we observe and the underlying micro-scale models we assume. Among the various applications benefiting from homogenization, Electric 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 cardio-pulmonary imaging, where current flow in fine-scale tissues underlies the resulting coarse-scale images. In this paper, we introduce a geometric approach for the homogenization (simulation) and inverse homogenization (imaging) of divergenceform elliptic operators with rough conductivity coefficients in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions over the domain. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of the domain when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. We explicitly give the transformations which map conductivity coefficients into divergencefree matrices and convex functions, as well as their respective inverses. Using weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain a robust homogenization algorithm for arbitrary rough coefficients, extending the global optimality of Delaunay triangulations with respect to a discrete Dirichlet energy to weighted Delaunay triangulations. We then consider inverse homogenization, which is known to be both non-linear and severely illposed, but that we can decompose into a linear ill-posed problem and a well-posed non-linear problem. Finally, our new geometric approach to homogenization and inverse homogenization is applied to EIT.