Ana Maria Soane

Andrei Draganescu, Ana Maria Soane

In this work we construct multigrid preconditioners to accelerate the solution process of a linear-quadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system (the KKT system) connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schur-complement of the block associated with the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.

Ana Maria Soane, Andrei Draganescu

The focus of this work is on the construction and analysis of optimal-order multigrid preconditioners to be used in the Newton-Krylov method for a distributed optimal control problem constrained by the stationary Navier-Stokes equations. As in our earlier work [7] on the optimal control of the stationary Stokes equations, the strategy is to eliminate the state and adjoint variables from the optimality system and solve the reduced nonlinear system in the control variables. While the construction of the preconditioners extends naturally the work in [7], the analysis shown in this paper presents a set of significant challenges that are rooted in the nonlinearity of the constraints. We also include numerical results that showcase the behavior of the proposed preconditioners and show that for low to moderate Reynolds numbers they can lead to significant drops in number of iterations and wall-clock savings.