On the convergence of right transforming iterations for the numerical solution of PDE constrained optimization problems

We present an iterative solver, called right transforming iterations (or right transformations), for linear systems with a certain structure in the system matrix, such as they typically arise in the framework of KKT conditions for optimization problems under PDE constraints. The construction of the right transforming scheme depends on an inner approximate solver for the underlying PDE subproblems. We give a rigorous convergence proof for the right transforming iterative scheme in dependence on the convergence properties of the inner solver. Provided that a fast subsolver is available, this iterative scheme represents an efficient way of solving first order optimality conditions. Numerical examples endorse the theoretically predicted contraction rates