Inexact and preconditioned Uzawa algorithms for saddle point problems
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners
SIAM Journal on Numerical Analysis
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
SIAM Journal on Numerical Analysis
Block-Triangular Preconditioners for Saddle Point Problems with a Penalty Term
SIAM Journal on Scientific Computing
A Note on Preconditioning for Indefinite Linear Systems
SIAM Journal on Scientific Computing
Analysis of iterative methods for saddle point problems: a unified approach
Mathematics of Computation
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
On nonsymmetric saddle point matrices that allow conjugate gradient iterations
Numerische Mathematik
Combination Preconditioning and the Bramble-Pasciak$^{+}$ Preconditioner
SIAM Journal on Matrix Analysis and Applications
Multigrid Methods for PDE Optimization
SIAM Review
Preconditioning Saddle-Point Systems with Applications in Optimization
SIAM Journal on Scientific Computing
Optimal Solvers for PDE-Constrained Optimization
SIAM Journal on Scientific Computing
A multigrid method for constrained optimal control problems
Journal of Computational and Applied Mathematics
SIAM Journal on Matrix Analysis and Applications
All-at-once solution of time-dependent Stokes control
Journal of Computational Physics
Computational Optimization and Applications
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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 nonstandard 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 we believe other problems could be treated in the same way. We give numerical results, and we compare these with those obtained by solving the equivalent forward problem using similar techniques.