In this work, we propose to efficiently solve time dependent parametrized optimal control problems governed by parabolic partial differential equations through the certified reduced basis method. In particular, we will exploit an error estimator procedure, based on easy-to-compute quantities which guarantee a rigorous and efficient bound for the error of the involved variables. First of all, we propose the analysis of the problem at hand, proving its well-posedness thanks to Nečas - Babuška theory for distributed and boundary controls in a space-time formulation. Then, we derive error estimators to apply a Greedy method during the offline stage, in order to perform, during the online stage, a Galerkin projection onto a low-dimensional space spanned by properly chosen high-fidelity solutions. We tested the error estimators on two model problems governed by a Graetz flow: a physical parametrized distributed optimal control problem and a boundary optimal control problem with physical and geometrical parameters. The results have been compared to a previously proposed bound, based on the exact computation of the Babuška inf-sup constant, in terms of reliability and computational costs. We remark that our findings still hold in the steady setting and we propose a brief insight also for this simpler formulation.
3D-VAR for Parametrized Partial Differential Equations: A Certified Reduced Basis Approach
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University（亚琛高等研究学院计算工程科学研究所（AICES），亚琛工业大学）；Numerical Mathematics (IGPM), RWTH Aachen University（数值数学（IGPM），亚琛工业大学）；Aachen Institute for Advanced Study in Computational Engineering Science (AICES) and Faculty of Civil Engineering, RWTH Aachen University（亚琛高等研究学院计算工程科学研究所（AICES）和土木工程学院，亚琛工业大学）
A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems
ENS Cachan/CNRS/Université Paris-Saclay（巴黎高等师范学校卡尚分校/CNRS/巴黎萨克雷大学）；Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS（巴黎东大学，多尺度建模与模拟实验室，MSME UMR 8208 CNRS）
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Reduced Basis A Posteriori Error Bounds for the Instationary Stokes Equations
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University（亚琛计算工程科学高等研究院）；Faculty of Civil Engineering, RWTH Aachen University（亚琛工业大学土木工程学院）
