Identification of Conductivity: Some Recent Results about a Composite Map (Identification for Control)

The identification the leading coefficient (conductivity) in a second order ordinary differential equation (ODE) or in an elliptic partial differential equation (PDE) is the prototype of a class of inverse problems, which are relevant to environmental modelling. Herewith only identification of position dependent conductivity from interior measurements of both potential and source term will be considered. The following will be dealt with. 1) Uniqueness conditions and stability estimates in the one dimensional (ODE) case. A unifying view will be provided over said properties, which are affected by the regularity of the Cauchy problem for the unknown conductivity. Moreover, a non local uniqueness condition will be presented, which admits straightforward physical interpretation. 2) Stability estimates for the composite identification - and - control map, which relates measured potential, source term data to the potential computed from another source term and the conductivity identified from the former data pair. 3) The connection between dynamical systems and iterative algorithms, which identify anisotropic conductivity in two spatial dimensions by minimizing the equation error cost function. Some properties of the related gradient flows will be outlined as well as some numerical results.