Academic website of Stefan Takacs

FWF Erwin Schrödinger project J3362: Abstract

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 (PDE-constrained 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 large-scale 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 block-preconditioner 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 block-system (all-at-once approach or one-shot 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 PDE-constrained 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 non-linear 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 non-linear problems. The linearizations may arise as subproblems to be solved in Newton-like methods. For these subproblems, robustness is an particular issue as additional parameters, like penalty parameters or active sets, appear.