Robust solvers for PDE-constrained optimization problems

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 the grid level. In this case the overall complexity is linear in the number of unknowns ("optimal convergence order"). For standard elliptic problems, an optimal convergence order can be achieved using hierarchical methods, like multigrid methods. To obtain a robust and optimal convergence order, 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. Some of the mentioned problems are linear, 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.

The scope of the project was set on the development (and the analysis) of fast solvers for linear systems arizing from the discretization of partial differential equations (PDEs), where the main focus was set onto the aspect of robustness. Such solvers are, for instance, of interest in physics or engineering sciences, where many aspects, like fluid ow or heat transfer are modeled by PDEs. If we want to simulate such a flow or heat transfer, we have to solve the PDE. This can be done using a computer and the finite element method (FEM). Using a FEM discretization, one obtains a large-scale but sparse system of linear equations, which needs to be solved.Before the project started, the principal investigator was working on solvers for control problems. There, one tries to adjust some outer influence, like heat sources, such that a certain state, like a heat distribution, is achieved as good as possible. This can be formulated as an optimization problem, where the PDE, that models the heat transfer, is a constraint. So, we call this PDE-constrained optimization. The first task during the project was the extension to Stokes ow equations, which are considered more challenging than the heat equation. The optimization problem depends on a parameter, which can be understood as regularization parameter, and the condition number of the linear system grows if the parameter is decreased towards zero. Typically, this causes the convergence rates of a solver to deteriorate, which makes the computations slower. A second parameter is introduced with the discretization: the grid size. Again, the condition number grows if the grid gets refined. The aspect of robustness means that we are interested in iterative methods, where the number of iterations needed to reach a desired accuracy goal is independent of those parameters. For standard problems, one obtains robustness in the grid size by using hierarchical methods, like multigrid solvers. The goal was to follow the multigrid idea and develop an algorithm for solving the Stokes control problem, which shows robustness both, the grid size and the regularization parameter. Such a method could be successfully developed and analyzed within the project. It turned out that, as for the control of the heat transfer, achieving robustness in the regularization parameter was more challenging than achieving robustness in the grid size.During the work on the project, the question of robustness of solvers was also raised in another context: in the area of isogeometric analysis (IgA). Here, unlike in the case of finite elements, the solution is not represented by piecewise linear functions, but with splines or similar functions. Splines are piecewise polynomials, which are chosen such that the overall function itself is very smooth. During the last years, IgA gained much interest, also in engineering sciences. Again, achieving robustness in the grid size was not a problem, however known approaches showed deteriorating convergence rates if larger polynomial degrees are chosen. Within the Erwin Schrödinger project, a major step forward could be achieved in this direction. First, an robust approximation error estimate for splines has been shown, which was up to the knowledge of the principal investigator unknown in approximation theory. Simultaneously, it was shown that most of the splines also satisfy a corresponding inverse estimate. Using the information that this inverse estimate is only violated by splines, representing certain boundary effects, we could develop using a boundary correction the first robust multigrid solver for IgA.