A new approximation of the Schur complement in preconditioners for PDE-constrained optimization.
The authors consider the optimal control subject to partial differential equations (PDE) constraints. Their focus is on the solution of the discretized first-order conditions which are of saddle point form. The PDE-constraint considered is given by the Poisson equation. The authors introduce various general preconditioners well-suited to precondition linear systems in saddle point form. They then introduce a new Schur-complement approximation as an extension of the work presented in an article by \textit{T. Rees} et al. [SIAM J. Sci. Comput. 32, No. 1, 271--298 (2010; Zbl 1208.49035)]. The authors construct their approximation in such a way that more terms of the original Schur-complement are matched in contrast to previously existing results. Then, using some basic facts about finite element matrices, they prove that the eigenvalues of the preconditioned Schur-complement are bounded in a tight interval independently of the mesh and regularization parameters. The authors proceed to perform numerical experiments indicating that the new approximation is more robust with respect to the regularization parameter.
scientific article; zbMATH DE number 6188617
scientific article