Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition
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| Format: | Preprint |
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2024
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| _version_ | 1866929648558931968 |
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| author | Breiten, Tobias Burela, Shubhaditya Schulze, Philipp |
| author_facet | Breiten, Tobias Burela, Shubhaditya Schulze, Philipp |
| contents | Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against the conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18950 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition Breiten, Tobias Burela, Shubhaditya Schulze, Philipp Optimization and Control Dynamical Systems Fluid Dynamics Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against the conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search. |
| title | Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition |
| topic | Optimization and Control Dynamical Systems Fluid Dynamics |
| url | https://arxiv.org/abs/2412.18950 |