PhD Defence: Efficient Iterative Solvers for Saddle Point Systems arising in PDE-constrained Optimization Problems with Inequality Constraints

Efficient Iterative Solvers for Saddle Point Systems arising in PDE-constrained Optimization Problems with Inequality Constraints

This talk deals with the construction and analysis of efficient solution methods for the following three optimal control problems, namely the distributed optimal control of elliptic equations, the distributed optimal control of multiharmonic-parabolic equations and the distributed optimal control of the Stokes equations, all with pointwise inequality constraints on the control or Moreau-Yosida regularized constraints on the state. These additional constraints render the resulting first-order optimality system nonlinear. For linearization a primal-dual active set method is applied. The resulting large scale saddle point system to be solved in each step of the active set method (after discretization) depends on various model and discretization parameters. Therefore, in order to obtain efficient solvers, appropriate preconditioners are needed, that improve the spectral properties of the systems with respect to the parameter-dependencies. The main focus of this talk is the construction of such efficient preconditioners for the three mentioned problem classes. Finally, numerical results demonstrate their performance.