Towards Matrix‐Free AD‐Based Preconditioning of KKT Systems in PDE‐Constrained Optimization

The presented approach aims at solving an equality constrained, finite‐dimensional optimization problem, where the constraints arise from the discretization of some partial differential equation (PDE) on a given space grid. For this purpose, a stationary point of the Lagrangian is computed using Newton's method, which requires the repeated solution of KKT systems. The proposed algorithm focuses on two topics: Firstly, Algorithmic Differentiation (AD) will be used to evaluate the necessary computations of gradients, Jacobian‐vector products, and Hessian‐vector products, so that only the objective f ( y , u ) and the PDE constraint e ( y , u ) = 0 have to be specified by the user. Secondly, we solve the KKT system iteratively using the QMR algorithm, with preconditioning provided by a multigrid approach. We wish to explore whether the Jacobian‐vector products provided by AD are sufficient to construct suitable multigrid preconditioners. Our approach is then embedded into a globalized optimization routine. Numerical results for optimization problems involving a nonlinear reaction‐diffusion model will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)