In this thesis, we will focus on inverse problems appearing in ultra precise turning processes. Ultra precision turning is widely used to manufacture metallic surfaces with high surface quality. One crucial influencing factor of the surface quality is unbalances leading to vibrations of the machine structure and interaction with the cutting process. This interaction is the so-called process machine interaction. Therefore, a model is built which simulates the influence of unbalances of the machine structure and process parameters on the resulting surface of the workpiece. In order to include the process machine interaction into the model, a new force model for ultra precision turning is developed. The resulting interaction model is based on a nonlinear parameter-dependent system of ordinary differential equations. The corresponding forward model is thus described by the map connecting the input parameters to the solution of this equation system. The main part of the thesis is the inversion of the forward operator, i.e. for a given tool path on the workpiece the necessary input parameters are computed such that solving the forward model with this new input parameters results in the desired tool path. Since the forward problem is ill-posed, regularization methods with sparsity constraints are applied which promote sparse solutions. The advantage of such sparse solutions is that they limit the points of machine changes in the machine control. Two different applications are treated in detail and illustrated with various numerical examples.