Multiobjective Optimal Control of Partial Differential Equations Using Reduced-Order Modeling
Description
The principal aim of this project is to develop efficient numerical techniques for the computation of the Pareto set of PDE-constrained multiobjective optimal control problems. This set encompasses the optimal compromises between several, typically conflicting objectives. The solution of the partial differential equations contained in the problem formulation requires a considerable numerical effort. Therefore, already the solution of a single-objective optimization problem often becomes a challenge, and the solution of a multiobjective optimization problem with PDE constraints quickly becomes practically infeasible. To solve this problem, in this project we will combine model reduction techniques for partial differential equations with algorithms for the set oriented solution of multiobjective optimization problems.
To develop efficient numerical methods, we will first analyse existing methods for the solution of multiobjective optimization problems with respect to their behavior in the presence of inexact function and derivative evaluations. Secondly, we will extend these algorithms to enable them to deal with reduced models. Usage of such models, which are derived on the basis of the well-known proper orthogonal decomposition (POD), introduces an approximation error that propagates into errors in further quantities that are essential for the application of the above-mentioned method. For this reason we will derive analytical bounds for the magnitudes of these errors which allow us to understand the dependence of the errors on parameters used in the model reduction procedure. Subsequently, this makes it possible to develop algorithms adapted to the considered problem class that utilize automatically chosen model reduction strategies.
We will first realize the above-sketched research program for a specific problem class. In the later stages of the project, more complex problem formulations will be considered that include e.g. state constraints or non-smooth objective functions.
Publications
Sebastian Peitz, Stefan Klus: Koopman operator-based model reduction for switched-system control of PDEs, Automatica 106, 184-191, 2019.
Sebastian Peitz, Sina Ober-Blöbaum, Michael Dellnitz: Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling, Acta Applicandae Mathematicae 161(1), 171-199, 2019.
Bennet Gebken, Sebastian Peitz, Michael Dellnitz: On the hierarchical structure of Pareto critical sets, Journal of Global Optimization 73(4), 891-913, 2019.
Sebastian Peitz, Michael Dellnitz: A Survey of Recent Trends in Multiobjective Optimal Control - Surrogate Models, Feedback Control and Objective Reduction, Math. Comput. Appl. 23(2), 2018.
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: Set-Oriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition, In: Keiper W., Milde A., Volkwein S. (eds) Reduced-Order Modeling (ROM) for Simulation and Optimization. p.47-72, Springer, 2018 (SPP1962-017).
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: Set-Oriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition , Winfried Keiper, Anja Milde, Stefan Volkwein (Eds.), "Reduced-Order Modeling (ROM) for Simulation and Optimization", 47-72, Springer., 2018 (SPP1962-017).
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: POD-Based Multiobjective Optimal Control of PDEs with Non-smooth Objectives , Proceedings in Applied Mathematics and Mechanics (PAMM) 17, 51-54, 2017 (SPP1962-024).
Stefan Banholzer, Eugen Makarov, Stefan Volkwein: POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena , European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), 881-888., 2017 (SPP1962-043).
Stefan Banholzer, Dennis Beermann, Stefan Volkwein: POD-Based Error Control for Reduced-Order Bicriterial PDE-Constrained Optimization , Annual Reviews in Control 44, 226-237., 2017 (SPP1962-015).
Stefan Banholzer, Dennis Beermann, Stefan Volkwein: POD-Based Bicriterial Optimal Control by the Reference Point Method, IFAC-PapersOnLine 49(8), 210-215, 2016 (SPP1962-014).
Preprints
Stefan Banholzer, Stefan Volkwein: Hierarchical Convex Multiobjective Optimization by the Euclidean Reference Point Method (SPP1962-117, 08/2019, [bib])
Stefan Banholzer, Eugen Makarov, Stefan Volkwein: POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena (SPP1962-043, 11/2017, [bib])
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: POD-Based Multiobjective Optimal Control of PDEs with Non-smooth Objectives (SPP1962-024, 06/2017, [bib])
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: Set-Oriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition (SPP1962-017, 04/2017, [bib])
Stefan Banholzer, Dennis Beermann, Stefan Volkwein: POD-Based Bicriterial Optimal Control by the Reference Point Method (SPP1962-014, 04/2017, [bib])
Stefan Banholzer, Dennis Beermann, Stefan Volkwein: POD-Based Error Control for Reduced-Order Bicriterial PDE-Constrained Optimization (SPP1962-015, 04/2017, [bib])
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