If there is a relationship between the sought parameters and observable
quantities in the form of a partial differential equation, such parameter
estimation (or inverse) problems can be cast as an optimization problem
involving PDEs as constraints. Unfortunately, inverse problems are
numerically very challenging, involving the solution of a large number of PDEs as subproblems. Adaptive finite element methods are therefore an important tool to reduce the amount of work, or to achieve otherwise unattainable accuracy. We will present error estimation techniques to generate discretizations for such problems that independently adapt both the meshes for the (observable) state variables as well as those for the sought parameters. The error estimates are based on residuals of the optimality conditions and are weighted with the solution of a dual problem. We will show results demonstrating that this yields significant savings in computational work, as well as much increased resolution and accuracy compared to traditional approaches.