Continuous flows, which identify distributed parameters

Consider a class of ordinary and partial differential (PDE) equations, where the leading coefficient, conductivity, is to be determined from the potential, source term pair. The problem is often met in applications e.g., geophysics, reservoir modelling, diffusion processes. Some algorithms, which identify conductivity by minimizing the equation error V are described, as well as their heuristic relation with non linear evolution PDEs. Two sufficient time decay laws for V are obtained. They correspond to two different gradient flows i.e., identification algorithms. One flow is Hamiltonian. The evolution PDE of other flow is simplified by one integration step and a relation with an auxiliary elliptic BV problem is established. The discrete time setting is considered. An unconstrained, one step minimization rule is presented.