POD Suboptimal Control of Evolution Problems: Theory and Applications
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911834627375104 |
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| author | Banholzer, Stefan Beermann, Dennis Mechelli, Luca Volkwein, Stefan |
| author_facet | Banholzer, Stefan Beermann, Dennis Mechelli, Luca Volkwein, Stefan |
| contents | The work is organized as follows. First an introduction is given in Chapter 1. In Chapter 2 we introduce the POD method in finite and infinite-dimensional Hilbert spaces and discuss various applications. Chapter 3 is devoted to to POD-based Galerkin schemes for evolution problems. Mainly, we study linear problems taking different discretization methods into account. We provide a certified a-priori and a-posteriori error analysis. Furthermore, the numerical realizations are explained and illustrated by test examples. Quadratic programming problems governed by liner evolution problems are investigated in Chapter 4. As in Chapter 3 we present a certified a-priori and a-posteriori error analysis. Moreover, we discuss basis update strategies. In Chapter 5 we give an outlook to further directions in reduced-order modeling in optimal control and optimization. More precisely, a nonlinear optimal control problem is studied. Moreover, state-constrained optimization problems are solved by a tailored combination of primal-dual active set methods and POD-aesed reduced-order modeling. Furthermore, POD Galerkin methods for multiobjective optimal control problems are investigated. Finally, some required theoretical foundations are summarized in the appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | POD Suboptimal Control of Evolution Problems: Theory and Applications Banholzer, Stefan Beermann, Dennis Mechelli, Luca Volkwein, Stefan Optimization and Control Numerical Analysis Analysis of PDEs The work is organized as follows. First an introduction is given in Chapter 1. In Chapter 2 we introduce the POD method in finite and infinite-dimensional Hilbert spaces and discuss various applications. Chapter 3 is devoted to to POD-based Galerkin schemes for evolution problems. Mainly, we study linear problems taking different discretization methods into account. We provide a certified a-priori and a-posteriori error analysis. Furthermore, the numerical realizations are explained and illustrated by test examples. Quadratic programming problems governed by liner evolution problems are investigated in Chapter 4. As in Chapter 3 we present a certified a-priori and a-posteriori error analysis. Moreover, we discuss basis update strategies. In Chapter 5 we give an outlook to further directions in reduced-order modeling in optimal control and optimization. More precisely, a nonlinear optimal control problem is studied. Moreover, state-constrained optimization problems are solved by a tailored combination of primal-dual active set methods and POD-aesed reduced-order modeling. Furthermore, POD Galerkin methods for multiobjective optimal control problems are investigated. Finally, some required theoretical foundations are summarized in the appendix. |
| title | POD Suboptimal Control of Evolution Problems: Theory and Applications |
| topic | Optimization and Control Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2404.07015 |