Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations

Solving problems regarding the optimal control of partial differential equations (PDEs) -- also known as PDE-constrained optimization -- is a frontier area of numerical analysis. Of particular interest is the problem of flow control, where one would like to effect some desired flow by exerting, for example, an external force. The bottleneck in many current algorithms is the solution of the optimality system -- a system of equations in saddle point form that is usually very large and ill-conditioned. In this paper we describe two preconditioners -- a block-diagonal preconditioner for the minimal residual method and a block-lower triangular preconditioner for a non-standard conjugate gradient method -- which can be effective when applied to such problems where the PDEs are the Stokes equations. We consider only distributed control here, although other problems -- for example boundary control -- could be treated in the same way. We give numerical results, and compare these with those obtained by solving the equivalent forward problem using similar techniques.