A posteriori error estimation for finite element discretization of parameter identification problems

In this paper we develop an a posteriori error estimator for parameter identification problems. The state equation is given by a partial differential equation involving a finite number of unknown parameters. The presented error estimator aims to control the error in the parameters due to discretization by finite elements. For this, we consider the general setting of a partial differential equation written in weak form with abstract parameter dependence. Exploiting the special structure of the parameter identification problem, allows us to derive an error estimator which is cheap in comparison to the overall optimization algorithm. Several examples illustrating the behavior of an adaptive mesh refinement algorithm based on our error estimator are discussed in the numerical section. For the problems considered here, both, the efficiency of the estimator and the quality of the generated meshes are satisfactory.