Multigrid solution of a distributed optimal control problem constrained by the Stokes equations

TL;DR

This paper develops multigrid preconditioners to efficiently solve a linear-quadratic optimal control problem constrained by the Stokes equations, significantly reducing iteration counts as resolution increases.

Contribution

It introduces a novel multigrid preconditioning approach for the Schur-complement system in Stokes-constrained optimal control problems, achieving optimal convergence properties.

Findings

Number of conjugate gradient iterations decreases with finer resolution.

Multigrid preconditioners are of optimal order for the Stokes system.

Efficient elimination of state and adjoint variables enhances solver performance.

Abstract

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.