A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization

Abstract

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

A trust region reduced basis Pascoletti-Serafini algorithm for multi-objective PDE-constrained parameter optimization

“…This procedure is followed, e.g., by the methods presented in [30,24,15,3]. Let us specify that in [24,15,3] the authors proposed and progressively improved an RB method combined with a TR algorithm, based on more general results presented in [30]. Such a method constructs the RB space adaptively while the optimizer is computing the optimal solution.…”

“…The trust regions are defined instead through the RB error estimator, which is in fact the way one should use to check the quality of the approximation. In [15] also the importance of introducing a correction term on the RB level is discussed to improve the performance of the method. We point out that this only has to be done if one chooses two separate RB spaces for state and adjoint equations (see also [3]).…”

“…Because of this fact, [24] proposed a sufficient (Line 5) and a necessary (Line 17) condition for the sufficient decrease condition. In [15] it is also noted that the full-order quantities J (u (j+1) ) and ∇J (u (j+1) ) Algorithm 2 TR-RB algorithm 1: Initialize the reduced-order model at u (0) , set j = 0 and Loop flag=True; 2: while Loop flag do 3:…”

“…Luckily, the focus has shifted recently towards adapting the RB space while proceeding with the optimization method. This procedure is followed, e.g., by the methods presented in [30,24,15,3]. Let us specify that in [24,15,3] the authors proposed and progressively improved an RB method combined with a TR algorithm, based on more general results presented in [30].…”

“…We propose an extension of the method in [1] for solving multi-objective PDE-constained parameter optimization problems. This procedure is based on a combination of a trustregion reduced basis method [3,15] and the PS method. In particular, we discuss different strategies to handle the increasing number of reduced basis functions, which is crucial in order to guarantee good performances of the algorithm.…”