Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linear-quadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linear-quadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that the performance of the two-level versions of our preconditioners is independent of the mesh size and of the subdomain size. Our numerical studies indicate that the performance of our preconditioners for optimal control problems is comparable to the performance of their counterparts applied to single PDEs. Moreover, the performance of our preconditioners seems to be rather insensitive to the size of the control regularization parameter