A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization

Keil T, Mechelli L, Ohlberger M, Schindler F, Volkwein S

Research article (journal) | Peer reviewed

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.

Details about the publication

Journal: ESAIM: Mathematical Modelling and Numerical Analysis

Volume: 55

Status: Published

Release year: 2021

Language in which the publication is written: English

DOI: 10.1051/m2an/2021019

Link to the full text: https://www.esaim-m2an.org/articles/m2an/abs/2021/04/m2an200123/m2an200123.html

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

Authors from the University of Münster

Keil, Tim	Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger) Institute for Analysis and Numerics
Ohlberger, Mario	Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger) Center for Nonlinear Science Center for Multiscale Theory and Computation (CMTC)
Schindler, Felix Tobias	Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger) Institute for Analysis and Numerics

Projects the publication originates from

Cluster of Excellence 2044 - Mathematics Münster: Dynamics – Geometry – Structure (EXC 2044) Duration: 01/01/2019 - 31/12/2025 \| 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Main DFG-project hosted at University of Münster
EXC 2044 - C4: Geometry-based modelling, approximation, and reduction Duration: 01/01/2019 - 31/12/2025 \| 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Subproject in DFG-joint project hosted at University of Münster
Localized Reduced Basis Methods for PDE-constrained Parameter Optimization (LRB-Opt) Duration: 01/01/2019 - 30/06/2023 Funded by: DFG - Individual Grants Programme Type of project: Individual project

Promotionen, aus denen die Publikation resultiert

Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization
Candidate: Keil, Tim | Supervisors: Ohlberger, Mario | Reviewers: Ohlberger, Mario; Volkwein, Stefan
Period of time: 01/03/2018 - 22/06/2022
Doctoral examination procedure finished at: Doctoral examination procedure at University of Münster