Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems

We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state equation of the partial differential equation defines an isomorphism onto $H^{1,1/2}_{0;0,}(Q)$. Thus, we can eliminate the control from the tracking type functional to be minimized, to derive the optimality system in order to determine the state. Since the appearing operator induces an equivalent norm in $H_{0;0,}^{1,1/2}(Q)$, we will replace it by a computable realization of the anisotropic Sobolev norm, using a modified Hilbert transformation. We are then able to link the cost or regularization parameter $\varrho>0$ to the distance of the state and the desired target, solely depending on the regularity of the target. For a conforming space-time finite element discretization, this behavior carries over to the discrete setting, leading to an optimal choice $\varrho = h_x^2$ of the regularization parameter $\varrho$ to the spatial finite element mesh size $h_x$. Using a space-time tensor product mesh, error estimates for the distance of the computable state to the desired target are derived. The main advantage of this new approach is, that applying sparse factorization techniques, a solver of optimal, i.e., almost linear, complexity is proposed and analyzed. The theoretical results are complemented by numerical examples, including discontinuous and less regular targets. Moreover, this approach can be applied also to optimal control problems subject to non-linear state equations.