The existence of thick triangulations -- an "elementary" proof

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also emphasized.Comment: 7 pages Short not

Electrical networks and Stephenson's conjecture

In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome

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

We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients $\sigma(x)$ 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 $\Omega$. 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 $\Omega$ when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it 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. It is known that 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 the space of convex functions is the natural space in which to parameterize solutions of the EIT problem