Robust solvers for PDE-constrained optimization problems
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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. [...]