A preconditioning technique for a class of PDE-constrained optimization problems
Advances in Computational Mathematics
SIAM Journal on Matrix Analysis and Applications
A Robust Multigrid Method for Elliptic Optimal Control Problems
SIAM Journal on Numerical Analysis
Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations
SIAM Journal on Scientific Computing
Preconditioning for Allen-Cahn variational inequalities with non-local constraints
Journal of Computational Physics
All-at-once solution of time-dependent Stokes control
Journal of Computational Physics
A radial basis function method for solving PDE-constrained optimization problems
Numerical Algorithms
An all-at-once approach for the optimal control of the unsteady Burgers equation
Journal of Computational and Applied Mathematics
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Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.