  In this contribution we device and analyze improved variants of the
non-conforming dual approach for trust-region reduced basis (TR-RB)
approximation of PDE-constrained parameter optimization that has recently been
introduced in [Keil et al.. A non-conforming dual approach for adaptive
Trust-Region Reduced Basis approximation of PDE-constrained optimization.
arXiv:2006.09297, 2020]. The proposed methods use model order reduction
techniques for parametrized PDEs to significantly reduce the computational
demand of parameter optimization with PDE constraints in the context of
large-scale or multi-scale applications. The adaptive TR approach allows to
localize the reduction with respect to the parameter space along the path of
optimization without wasting unnecessary resources in an offline phase. The
improved variants employ projected Newton methods to solve the local
optimization problems within each TR step to benefit from high convergence
rates. This implies new strategies in constructing the RB spaces, together with
an estimate for the approximation of the hessian. Moreover, we present a new
proof of convergence of the TR-RB method based on infinite-dimensional
arguments, not restricted to the particular case of an RB approximation and
provide an a posteriori error estimate for the approximation of the optimal
parameter. Numerical experiments demonstrate the efficiency of the proposed
methods.
