POD-based error control for reduced-order bicriterial PDE-constrained optimization

Banholzer, Stefan; Beermann, Dennis; Volkwein, Stefan

doi:10.1016/j.arcontrol.2017.09.004

Cited by 15 publications

(19 citation statements)

References 26 publications

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“…Figure 7b-c for a heat flow problem with a tracking and a cost minimization objective). The third ROM approach was used in [108,109]. The difficulty here is that the minimization of the distance to the target point results in a more complicated objective function, which has to be treated carefully.…”

Section: Scalarizationmentioning

confidence: 99%

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A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

Peitz

Dellnitz

2018

MCA

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Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

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Section: Scalarizationmentioning

confidence: 99%

“…This way, also non-convex problems can be treated. In fact, due to the reference point method, the objective function is always convex [109], which can be exploited during the optimization. Alternative scalarization methods are the -constraint method [136] or lexicographic ordering [137].…”

Section: Online Multiobjective Optimizationmentioning

confidence: 99%

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

Peitz

Dellnitz

2018

MCA

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“…In this paper we apply the reference point method [11] in order to transform a bicriterial optimal control problem into a sequence of scalar-valued optimal control problems and solve them using well-known optimal control techniques; see [13]. We build on and extend previous results obtained in [2], where a linear convection-diffusion equation was considered. In addition, we allow the convection term to be time-dependent here.…”

Section: Introductionmentioning

confidence: 99%

POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena

Banholzer

Makarov

Volkwein

2019

Lecture Notes in Computational Science and Engineering

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In the present paper, a multiobjective optimal control problem governed by a heat equation with time-dependent convection term and bilateral control constraints is considered. For computing Pareto optimal points and approximating the Pareto front, the reference point method is applied. As this method transforms the multiobjective optimal control problem into a series of scalar optimization problems, the method of proper orthogonal decomposition (POD) is introduced as an approach for model-order reduction. New strategies for efficiently updating the POD basis in the optimization process are proposed and tested numerically.

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“…This way, also non-convex problems can be treated. In fact, due to the reference point method, the objective function is always convex [109] which can be exploited during the optimization. Alternative scalarization methods are the -constraint method [136] or lexocographic ordering [137].…”

Section: Online Multiobjective Optimizationmentioning

confidence: 99%

“…Figure 7 (b)-(c) for a heat flow problem with a tracking and a cost minimization objective). The third ROM approach was used in [108,109]. The difficulty here is that the minimization of the distance to the target point results in a more complicated objective function which has to be treated carefully.…”

Section: Scalarizationmentioning

confidence: 99%

A Survey of Recent Trends in Multiobjective Optimal Control -- Surrogate Models, Feedback Control and Objective Reduction

Peitz¹,

Dellnitz²

2018

Preprint

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Abstract:Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto optimal solutions have led to a wide range of new applications related to optimal and feedback control which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview over recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

show abstract

POD-based error control for reduced-order bicriterial PDE-constrained optimization

Cited by 15 publications

References 26 publications

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena

A Survey of Recent Trends in Multiobjective Optimal Control -- Surrogate Models, Feedback Control and Objective Reduction

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