### Abstract or Additional Information

Since the 1970s, multigrid methods have become ubiquitous in solving large-scale linear and nonlinear systems representing discretizations of partial differential equations (PDEs). The multigrid paradigm is that such systems can be solved efficiently by taking advantage of multiple resolutions of the PDE during the solution process. Instead, the systems arising in optimal control problems constrained by PDEs - a research area in which the scientific and engineering communities have shown an increased interest more recently - have different properties than those arising in discrete PDEs, and they require a different treatment.

In this talk we present new embodiments of the multigrid paradigm, which, in our view, are more appropriate for optimal control problems constrained by PDEs; in particular, the multigrid methods in question provide tools for solving such problems at a cost that is decreasing with increasing resolution relative to the cost of solving the associated PDEs. The focus will be on recent results on optimal order multigrid preconditioners for linear systems arising in the semismooth Newton solution process of certain control-constrained problems.