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In this contribution we device and analyze improved variants of thenon-conforming dual approach for trust-region reduced basis (TR-RB)approximation of PDE-constrained parameter optimization that has recently beenintroduced in [Keil et al.. A non-conforming dual approach for adaptiveTrust-Region Reduced Basis approximation of PDE-constrained optimization.arXiv:2006.09297, 2020]. The proposed methods use model order reductiontechniques for parametrized PDEs to significantly reduce the computationaldemand of parameter optimization with PDE constraints in the context oflarge-scale or multi-scale applications. The adaptive TR approach allows tolocalize the reduction with respect to the parameter space along the path ofoptimization without wasting unnecessary resources in an offline phase. Theimproved variants employ projected Newton methods to solve the localoptimization problems within each TR step to benefit from high convergencerates. This implies new strategies in constructing the RB spaces, together withan estimate for the approximation of the hessian. Moreover, we present a newproof of convergence of the TR-RB method based on infinite-dimensionalarguments, not restricted to the particular case of an RB approximation andprovide an a posteriori error estimate for the approximation of the optimalparameter. Numerical experiments demonstrate the efficiency of the proposedmethods.

An adaptive projected Newton non-conforming dual approach for trust-region reduced basis approximation of PDE-constrained parameter optimization