The main purpose of this project is to design efficient methods for imaging certain physical quantities from indirect measurements. The problems that we study arise in applications such as finding the electrical conductivity in the human body from electrical measurements on the skin, finding the elastic properties of tissue from controlled vibrations of the skin or even monitoring a pipeline using electrical measurements. The methods we design are efficient because they are based on networks that mimic the underlying physics of the problem. For example, to image the electrical conductivity we use a network of resistors.
The inverse problems on networks consist on using boundary data (analogous to the Dirichlet to Neumann map in continuum inverse problems) to find certain (scalar or vector valued) node quantities and (scalar or matrix valued) edge quantities defined on a known graph. We use ideas from Complex Geometric Optics to conjecture that if the linearization of such a problem is solvable, then the non-linear problem is also solvable. This would give a relatively simple way of checking solvability of such discrete inverse problems. Solvability results and reconstruction algorithms on networks are then used to solve a continuum inverse problem by interpreting the network problem as a discretization of the governing partial differential equation.
