On the Approximate Identification of the Leading Coefficient of a Linear Elliptic Equation by Output Error and Equation Error Minimization

By definition, the approximate solution to the (finite-dimensional) identification problem is a minimizing element of either cost function. The output error is minimized by two methods: finite difference-discretized gradient (FD-DZG) and finite element—discrete gradient (FE-DG). The equation error is minimized by a DZG method. Two-sided constraints are applied. With reference to an example of polynomial type in a rectangular domain, some tests are carried out in which the role of several parameters, including data noise, is investigated. Most computational results pertaining to the output error are interpreted according to: (a) sensitivity analysis; (b) an error estimate, which compares FD-DZG to FD-DG; and (c) a dynamical system model of the algorithm. An optimal uniform initial value of the unknown coefficient is found which yields a computed solution very close to the reference solution, regardless of domain discretization and output error minimization method. The superior noise immunity of FE-DG over FD-DZG is demonstrated. The results obtained from equation error minimization are less satisfactory.