---
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.

    **Keywords**: PDE constrained optimization, Trust-Region method, error analysis, Reduced Basis method, model order reduction, parametrized systems, large scale problems

    **AMS Mathematics Subject Classification**: 49M20, 49K20, 35J20, 65N30, 90C06
author:
- |
    Tim Keil$^\dag$, Luca Mechelli$^\ddag$, Mario Ohlberger$^\dag$,\
    Felix Schindler[^1], Stefan Volkwein[^2]
bibliography:
- 'bibliography.bib'
title: 'A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization[^3] '
---

Introduction {#sec:introduction .unnumbered}
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We are concerned with the development and rigorous analysis of novel 