Multiobjective Optimal Control of Partial Differential Equations Using ReducedOrder Modeling
Description
The principal aim of this project is to develop efficient numerical techniques for the computation of the Pareto set of PDEconstrained 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 singleobjective 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 wellknown proper orthogonal decomposition (POD), introduces an approximation error that propagates into errors in further quantities that are essential for the application of the abovementioned 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 abovesketched 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 nonsmooth objective functions.
Publications
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: SetOriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition, In: Keiper W., Milde A., Volkwein S. (eds) ReducedOrder Modeling (ROM) for Simulation and Optimization. p.4772, Springer, 2018 (SPP1962017).
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: PODBased Multiobjective Optimal Control of PDEs with Nonsmooth Objectives , Proceedings in Applied Mathematics and Mechanics (PAMM) 17, 5154, 2017 (SPP1962024).
Preprints
Stefan Banholzer, Stefan Volkwein: Hierarchical Convex Multiobjective Optimization by the Euclidean Reference Point Method (SPP1962117, 08/2019, [bib])
Stefan Banholzer, Eugen Makarov, Stefan Volkwein: PODBased Multiobjective Optimal Control of TimeVariant Heat Phenomena (SPP1962043, 11/2017, [bib])
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: PODBased Multiobjective Optimal Control of PDEs with Nonsmooth Objectives (SPP1962024, 06/2017, [bib])
Dennis Beermann, Michael Dellnitz, Sebastian Peitz, Stefan Volkwein: SetOriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition (SPP1962017, 04/2017, [bib])
Stefan Banholzer, Dennis Beermann, Stefan Volkwein: PODBased Bicriterial Optimal Control by the Reference Point Method (SPP1962014, 04/2017, [bib])
Stefan Banholzer, Dennis Beermann, Stefan Volkwein: PODBased Error Control for ReducedOrder Bicriterial PDEConstrained Optimization (SPP1962015, 04/2017, [bib])
Members
Project Related News

Aug 15, 2019 : New preprint submitted
Stefan Banholzer submitted the preprint SPP1962117, Hierarchical Convex Multiobjective Optimization by the Euclidean Reference Point Method

Apr 12, 2017 : Welcome to our new project member
Stefan Bauholzer joins project 5.