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

¹ Mathematics Münster, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
² Department of Mathematics and Statistics, University of Konstanz, D-78457 Konstanz, Germany

^* Corresponding author: tim.keil@wwu.de

Received: 16 June 2020
Accepted: 12 April 2021

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.