Title  Robust solvers for PDEconstrained optimization problems 
Funding agency  Austrian Science Fund (FWF) 
Project number  J3362 
Principal investigator  Stefan Takacs 
Approval date  1 Oct 2012 
First research stay 
Andy Wathen Mathematical Institute University of Oxford Start: 15 Nov 2012 End: 16 Nov 2013 
Second research stay 
Roland Herzog Fakultät für Mathematik TU Chemnitz Start: 18 Nov 2013 End: 26 Nov 2014 
Return phase 
Ulrich Langer Department for Computational Mathematics (NUMA) Universität Linz Start: 1 Dec 2014 End: 30 Nov 2015 
Keywords 

The main scope of the proposed project is the construction and the analysis of fast numerical methods for solving optimization problems constrained to partial differential equations (PDEconstrained optimization problems). To this class of problems belong optimal control problems, optimal design problems, shape and topology optimization problems, and many others. We will concentrate on optimal control problems. The proposer has considered in his previous work elliptic partial differential equations (PDEs) as constraints. The solution of such a problem is characterized by the optimality system, which is a system of PDEs. Such systems can be discretized using standard techniques. The resulting system is typically a largescale linear system, which is sparse and indefinite. The construction of fast iterative solvers for such a system is of particular interest.
One issue in solving such linear systems is robustness. The optimal control problem itself depends on a parameter, which can be interpreted as regularization parameter or as cost parameter. If this parameter approaches zero, the condition number of the problem grows. Typically this leads to deteriorating convergence rates. A second parameter is introduced by discretizing the problem: the grid size or the grid level. Also here, if the grid is refined, the condition number of the problems grows. We are interested in an iterative method where the number of iterations is independent of these parameters. For standard elliptic problems, robustness in the grid parameter (optimal convergence order) can be achieved using hierarchical methods, like multigrid methods.
To obtain such a behavior also for an optimal control problem, several approaches are available. One possibility is the use of standard multigrid methods as a part of a blockpreconditioner that is applied in the framework of the outer iteration, like a Krylov space method. An alternative is to apply the multigrid idea to the whole blocksystem (allatonce approach or oneshot approach).
Both kinds of approaches have already been successfully applied to the elliptic optimal control problem mentioned above. The goal of the project is to develop such methods also for other PDEconstrained optimization problems, in particular for other optimal control problems. Such problems involve other state equations, like the Stokes control problem (velocity tracking), optimization of elastic deformation or optimization of Maxwell's equations. Also the consideration of box constraints for state or control, is of interest. The mentioned problems are linear but the constrained problems are nonlinear due to the constraints. The methods introduced above have the potential to solve, both, the problems problems which are already linear and the linearizations of the nonlinear problems. The linearizations may arise as subproblems to be solved in Newtonlike methods. For these subproblems, robustness is an particular issue as additional parameters, like penalty parameters or active sets, appear.