Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs.
The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves.
We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides
1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and
2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid
such that global convergence to the solution of the infinite-dimensional problem is ensured.
We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems.
Numerical results are presented that illustrate the potential of the approach.
This is joint work with Jan Carsten Ziems.