2009 SCAIM Seminar - 07

We consider homogenization of the operator $u \mapsto -\text{div} (\sigma \nabla u)$, which is parameterized by symmetric $\sigma \in \real^{2\times 2}$ for the special case where the domain $\Omega \subset \real^2$. We refer to $\sigma$ as the conductivity owing to is physical interpretation, and, in addition to conditions maintaining ellpticity, require only that $\sigma \in L^{\infty}$---that is, $\sigma$ may vary on a continuum of scales. This is in contrast to classical homogenization, which typically requires that rapid variations in $\sigma$ occur at length scales far below the dimensions of $\Omega$. Following work of Owhadi and Zhang (2007, 2008), we represent $\sigma$ in particular coordinates $F : \Omega \rightarrow \Omega$ such that its push-forward $Q = F_{\ast} \sigma$ is a divergence-free tensor. In $\real^2$, the consequence of the divergence-free constraint is that we can represent $Q$ using a scalar function $s(x)$. We show that sampling $s(x)$ at a coarse scale is equivalent to homogenizing $\sigma$ at that scale. For example, the effective anisotropy of a laminated material predicted by classical homogenization theory is manifested when $s(x)$ is sampled at a scale coarser than the pitch of laminations. We discuss two applications of this new parameterization of conductivity: 1) We present the efficient computation of triangulations well-adapted to the effective anisotropy of $\sigma$, in turn developing a new geometric understanding of weighted Delaunay triangulations; and 2) We show how our parameterization can play a role in Electric Impedence Tomography, an inverse method which {\em determines} $\sigma$ from boundary Dirichlet and Neumann data.