| Methods are proposed for the numerical solution of optimal control problems with partial differential equation (PDE) constraints and inequality constraints on the control variable. State and control variables are discretized using an adaptive finite-element approach. Algorithms for optimization and PDEs are combined to solve a discretized optimization problem over a sequence of adaptive meshes.; A primal interior-point method is used for the optimization part of the algorithm. In the PDE-context, the usually preferred primal-dual methods may have numerical difficulties if the dual variables are not sufficiently smooth. It is shown that these difficulties may be avoided if the primal method is implemented using an extrapolation scheme for the parameter mu.; The linear systems to be solved at each iteration are large, symmetric and have PDE-like structure. They become increasingly ill-conditioned as the solution is approached. In order to handle the size, sparsity and condition of these systems, a modified Newton method is used. The aim is to incorporate as much information as possible into the method while the cost of the computation low. To this end, the system matrix of the modified Newton method has the same block structure as the original system matrix and existing solvers are used for some of the blocks. In particular, an algebraic multigrid preconditioner with ILU smoothing is used for the PDE constraint blocks, and a symmetric Gauss-Seidel preconditioner is used for the control block and the full linear system. The preconditioner is fully parallel.; The PDE part of the algorithm uses adaptive mesh refinement based on an a posteriori hierarchical basis error estimator for the state variables. The path-following parameter mu and the PDE parameters are chosen to allow the discretization error and optimization error to go to zero at the same rate. An error estimator based on the state variable allows the mesh to be adaptively refined and unrefined without the additional cost of solving the adjoint equation.; These ideas are illustrated in the context of the elliptic finite-element PDE package PLTMG. Numerical results are presented. |
