Discrete Geometric Structures in Homogenization and Inverse Homogenization

We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions s(x) over the domain Ω. 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 Ω 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 divergence-free matrices and convex functions, as well as their respective inverses. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain an optimally 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. Next, we consider inverse homogenization, that is, the recovery of the microstructure from macroscopic information, a problem which is known to be both non-linear and severly ill-posed. We show how to decompose this reconstruction into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary Dirichlet-to-Neumann map. We also show that the EIT problem admits a unique solution in the space of divergence-free matrices. This is of practical importance since the medium to be recovered in practise may not be isotropic and the associated EIT problem may not admit an isotropic solution. As such, we suggest that the space of divergence-free matrices is the natural space in which to parameterize solutions of the EIT problem. ∗California Institute of Technology, MC 217-50 Pasadena, CA 91125, USA. [email protected], [email protected], [email protected] 1 ar X iv :0 90 4. 26 01 v1 [ m at h. A P] 1 7 A pr 2 00 9