Model order reduction for optimality systems through empirical gramians
In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE)...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2023-12-01
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| Series: | Frontiers in Applied Mathematics and Statistics |
| Subjects: | |
| Online Access: | https://www.frontiersin.org/articles/10.3389/fams.2023.1144142/full |
| _version_ | 1797397174174089216 |
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| author | Luca Mechelli Jan Rohleff Stefan Volkwein |
| author_facet | Luca Mechelli Jan Rohleff Stefan Volkwein |
| author_sort | Luca Mechelli |
| collection | DOAJ |
| description | In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples. |
| first_indexed | 2024-03-09T01:06:10Z |
| format | Article |
| id | doaj.art-e5fe60d67c224468a79011e6df07ab40 |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-03-09T01:06:10Z |
| publishDate | 2023-12-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-e5fe60d67c224468a79011e6df07ab402023-12-11T09:56:41ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872023-12-01910.3389/fams.2023.11441421144142Model order reduction for optimality systems through empirical gramiansLuca MechelliJan RohleffStefan VolkweinIn the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples.https://www.frontiersin.org/articles/10.3389/fams.2023.1144142/fullmodel order reductionempirical gramiansproper orthogonal decompositionparabolic partial differential equationsmultiobjective optimizationmodel predictive control |
| spellingShingle | Luca Mechelli Jan Rohleff Stefan Volkwein Model order reduction for optimality systems through empirical gramians Frontiers in Applied Mathematics and Statistics model order reduction empirical gramians proper orthogonal decomposition parabolic partial differential equations multiobjective optimization model predictive control |
| title | Model order reduction for optimality systems through empirical gramians |
| title_full | Model order reduction for optimality systems through empirical gramians |
| title_fullStr | Model order reduction for optimality systems through empirical gramians |
| title_full_unstemmed | Model order reduction for optimality systems through empirical gramians |
| title_short | Model order reduction for optimality systems through empirical gramians |
| title_sort | model order reduction for optimality systems through empirical gramians |
| topic | model order reduction empirical gramians proper orthogonal decomposition parabolic partial differential equations multiobjective optimization model predictive control |
| url | https://www.frontiersin.org/articles/10.3389/fams.2023.1144142/full |
| work_keys_str_mv | AT lucamechelli modelorderreductionforoptimalitysystemsthroughempiricalgramians AT janrohleff modelorderreductionforoptimalitysystemsthroughempiricalgramians AT stefanvolkwein modelorderreductionforoptimalitysystemsthroughempiricalgramians |