Trust region methods with hierarchical finite element models for PDE-constrained optimization

In this paper, a Hierarchical Trust Region Algorithm for solving PDE-constrained optimization problems is developed. A hierarchy of finite element meshes is used to define a hierarchy of quadratic models for the approximation of the discrete reduced cost functional on the finest mesh. The proposed algorithm simultaneously controls the choice of the model and the size of the trust region radius. Application of the trust region convergence theory allows for proving that every accumulation point of the sequence produced by the algorithm is a stationary point of the discretized problem. Numerical examples illustrate the behavior of the method and show a considerable reduction of computation time compared to the standard Newton trust region scheme.