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We introduce a new geometric approach for the homogenization and inversehomogenization of the divergence form elliptic operator with rough conductivitycoefficients $\sigma(x)$ in dimension two. We show that conductivitycoefficients are in one-to-one correspondence with divergence-free matrices andconvex functions $s(x)$ over the domain $\Omega$. Although homogenization is anon-linear and non-injective operator when applied directly to conductivitycoefficients, homogenization becomes a linear interpolation operator overtriangulations of $\Omega$ when re-expressed using convex functions, and is avolume averaging operator when re-expressed with divergence-free matrices.Using optimal weighted Delaunay triangulations for linearly interpolatingconvex functions, we obtain an optimally robust homogenization algorithm forarbitrary rough coefficients. Next, we consider inverse homogenization and showhow to decompose it into a linear ill-posed problem and a well-posed non-linearproblem. We apply this new geometric approach to Electrical ImpedanceTomography (EIT). It is known that the EIT problem admits at most one isotropicsolution. If an isotropic solution exists, we show how to compute it from anyconductivity having the same boundary Dirichlet-to-Neumann map. It is knownthat the EIT problem admits a unique (stable with respect to $G$-convergence)solution in the space of divergence-free matrices. As such we suggest that thespace of convex functions is the natural space in which to parameterizesolutions of the EIT problem.

Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT