In this contribution we propose and rigorously analyze new variants of adaptive Trust- Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non- conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly

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 multiscale 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.

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We are concerned with employing Model Order Reduction (MOR) to efficiently solve parameterized multiscale problems using the Localized Orthogonal Decomposition (LOD) multiscale method. Like many multiscale methods, the LOD follows the idea of separating the problem into localized fine-scale subproblems and an effective coarse-scale system derived from the solutions of the local problems. While the Reduced Basis (RB) method has already been used to speed up the solution of the fine-scale problems, the resulting coarse system remained untouched, thus limiting the achievable speed up. In this work we address this issue by applying the RB methodology to a new two-scale formulation of the LOD. By reducing the entire two-scale system, this two-scale Reduced Basis LOD (TSRBLOD) approach, yields reduced order models that are completely independent from the size of the coarse mesh of the multiscale approach, allowing an efficient approximation of the solutions of parameterized multiscale problems even for very large domains. A rigorous and efficient a posteriori estimator bounds the model reduction error, taking into account the approximation error for both the local fine-scale problems and the global coarse-scale system.

Projection based model order reduction has become a mature technique for simulation of large classes of parameterized systems. However, several challenges remain for problems where the solution manifold of the parameterized system cannot be well approximated by linear subspaces. While the online efficiency of these model reduction methods is very convincing for problems with a rapid decay of the Kolmogorov n-width, there are still major drawbacks and limitations. Most importantly, the construction of the reduced system in the offline phase is extremely CPU-time and memory consuming for large scale and multi scale systems. For practical applications, it is thus necessary to derive model reduction techniques that do not rely on a classical offline/online splitting but allow for more flexibility in the usage of computational resources. A promising approach with this respect is model reduction with adaptive enrichment. In this contribution we investigate Petrov-Galerkin based model reduction with adaptive basis enrichment within a Trust Region approach for the solution of multi scale and large scale PDE constrained parameter optimization.