A Substitution Method Applied to the Identification of the Leading Coefficient Appearing in Linear-Equations of Elliptic Type

The substitution method we describe consists of expressing the unknown coefficient as a function of data, of the solution to an auxiliary direct problem and of a control parameter. We discuss the connection between this method and equation error minimisation. The equations where the coefficient appears correspond to the following direct problems: (1) the two-point boundary value problem for an ordinary differential equation of elliptic type; (2) its finite-dimensional (discretised) counterpart, where also the domain is discretised; (3) an algebraic equation on a two-dimensional discrete domain, obtained from a Dirichlet boundary value problem. Uniqueness, stability and existence properties are considered. The last case is discussed with reference to a preliminary example, which shows the connection between the identification algorithm and a nonlinear discrete time dynamical system. For reasons of space, only hints for proofs are given.''