Comment on A.-P. Calderón’s paper: “On an inverse boundary value problem” [in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980; MR0590275 (81k:35160)]

Abstract:

Calderón determined a method to approximate the conductivity $\sigma$ of a conducting body in ${R^n}$ (for $n \geq 2$) based on measurements of boundary data. The approximation is good in the ${L_\infty }$ norm provided that the conductivity is a small perturbation from a constant. We calculate the approximation exactly for the case of homogeneous concentric conducting disks in ${R^2}$ with different conductivities. Here, the difference in the conductivities is the perturbation. We show that the approximation yields precise information about the spatial variation of $\sigma$, even when the perturbation is large. This ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications.