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 nonlinear 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 now 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