A differential formulation based on a perturbative approach to solve the ECT inverse problem Original Research Article

We describe a 3D numerical procedure for the reconstruction of conductivity profiles, a classic nonlinear inverse problem. The related direct field problem is discretized using a finite elements differential formulation in terms of magnetic vector potential and electric scalar potential. Adopting shape functions based on edge-elements, the vector potential is gauged using a tree—cotree decomposition of the finite element mesh. The nonlinear inverse problem is approximated by expanding, up to the second order, the nonlinear operator mapping the conductivity into the magnetic field at the probes location. The solution is obtained by minimizing an error functional related to the distance between measurements and their numerical approximation. The geometrical properties of this error functional are reviewed, showing how the presence of local minima can be controlled. From the numerical point of view, this perturbative approach enlarges the range of validity of the classic linear Born approximation, since the quadratic term takes into account, at least partially, the effects due to the reaction field. Several examples highlight the main features of the method.