Christian Grossmann

The quadratic loss penalty is a well known technique for optimization and control problems to treat constraints. In the present paper they are applied to handle control bounds in a boundary control problems with semilinear elliptic state equations. Unlike in the case of finite dimensional optimization for infinite dimensional problems the order of convergence could only be roughly estimated, but numerical experiments revealed a clearly better convergence behavior with constants independent of the dimension of the used discretization. The main result in the present paper is the proof of sharp convergence bounds for both, the finite und infinite dimensional problem with a mesh-independence in case of the discretization. Further, to achieve an efficient realization of penalty methods the principle of control reduction is applied, i.e. the control variable is represented by the adjoint state variable by means of
some nonlinear function. The resulting optimality system this way depends only on the state and adjoint state. This system is discretized by conforming linear finite elements. Numerical experiments show exactly the theoretically predicted behavior of the studied penalty technique.

The paper deals with the numerical treatment of the optimal control of drying of materials which may lead to cracks. The drying process is controlled by temperature, velocity and humidity of the surrounding air. The state equations dene the humidity and temperature distribution within a simplied wood specimen for given controls. The elasticity equation describes the internal stresses under humidity and temperature changes. To avoid cracks these internal stresses have to be limited. The related constraints are treated by smoothed exact barrier-penalty techniques. The objective functional of the optimal control problem is of tracking type. Further it contains a quadratic regularization by an energy term for the control variables (surrounding air) and barrier-penalty terms.
The necessary optimality conditions of the auxiliary problem form a coupled system of nonlinear equations in appropriate function spaces. This optimality system is given by the state equations and the related adjoint equations, but also by an approximate projection onto the admissible set of controls by means of barrier-penalty terms. This system is discretized by nite elements and treated iteratively for given controls. The optimal control itself is performed
by quasi-Newton techniques.

In this paper basic mathematical tasks of coordinate measurement are briefly described and a modied optimization algorithm is proposed. Coordinate measurement devices generate huge data set and require adapted methods to solve related mathematical problems in real time. The proposed algorithm possesses a simplied step size rule and nds the solution of the minimum circumscribed ball fitting after only a nite number The iteration is of the steepest descent type applied to the related distance function. But, in contrast to standard algorithms it uses a modied step size rule that takes into account the specic properties of the occurring objective function. This small dierence in the code improves the performance of the algorithm and it enables real time use of the proposed method in coordinate measurement machines. The eciency of the prosed algorithm will be illustrated by some typical examples.