A primal interior point method for control constrained optimal control problems
with PDE constraints is considered. Pointwise elimination of the control
leads to a homotopy in the remaining state and dual variables, which is addressed
by a short step pathfollowing method. The algorithm is applied to the
continuous, infinite dimensional problem, where discretization is performed
only in the innermost loop when solving linear equations. The a priori elimination
of the least regular control permits to obtain the required accuracy with
comparatively coarse meshes. Convergence of the method and discretization
errors are studied, and the method is illustrated at two numerical examples.
The paper introduces an identification problem arising in modern regional hyperthermia, a cancer
therapy aiming at heating the tumor by microwave radiation. The task is to identify the highly
individual perfusion, which affects the resulting temperature distribution, from MR measurements.
The identification problem is formulated as an optimization problem. Existence of a solution and
optimality conditions are analyzed. Different regularizations and problem variants are considered. For
the numerical solution, a standard SQP method is used. Sufficient conditions for the convergence of
the method are derived. Finally, numerical examples on artificial as well as clinical data are presented.