An important question is to non-invasively find the volume of each phase in body, by only probing its response at the boundary. Here we consider a body containing two phases arranged in any configuration, and address the inverse problem of bounding the volume fraction of each phase from electrical tomography measurements at the boundary, i.e. measurements of the current flux through the boundary produced by potentials applied at the boundary. It turns out that this problem is closely related to the extensively studied problem of bounding the effective conductivity of periodic composite materials. Those bounds can be used to bound the response of an arbitrarily shaped body, and if this response has been measured, they can be used to extract information about the volume fraction.

Numerical experiments show that for a wide range of inclusion shapes one of the bounds turns out to be close to the actual volume fraction. The bounds extend those obtained by Capdeboscq and Vogelius for asymptotically small inclusions. The same ideas can be extended to elasticity and used to incorporate thermal measurements as well as electrical measurements. The translation method for obtaining bounds on the effective conductivity can also be applied directly to bound the volume fraction of inclusions in a body. This is joint work with Hyeonbae Kang and Eunjoo Kim.