Simulation-Based Model Reduction for Partial Differential Equations on Networks
In this thesis, we consider model reduction for parameter dependent parabolic PDEs defined on networks with variable composition. For this type of problem, the Reduced Basis Element Method (RBEM), developed by Maday and Rønquist, is a reasonable choice as a solution on the entire domain is not requi...
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| Fformat: | Online |
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FAU University Press
2025
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| Mynediad Ar-lein: | ONIX_20251215T160703_9783961471560_24 |
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Dim Tagiau, Byddwch y cyntaf i dagio'r cofnod hwn!
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| _version_ | 1869527660008308736 |
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| author | Walther, Maximilian |
| author_browse | Walther, Maximilian |
| author_facet | Walther, Maximilian |
| author_sort | Walther, Maximilian |
| collection | Directory of Open Access Books |
| description | In this thesis, we consider model reduction for parameter dependent parabolic PDEs defined on networks with variable composition. For this type of problem, the Reduced Basis Element Method (RBEM), developed by Maday and Rønquist, is a reasonable choice as a solution on the entire domain is not required. The reduction method is based on the idea of constructing a reduced basis for every individual component and coupling the reduced elements using a mortar-like method. However, this decomposition procedure can lead to difficulties, especially for networks consisting of numerous edges. Due to the variable composition of the networks, the solution on the interfaces is extremely difficult to predict. This can lead to unsuitable basis functions and poor approximations of the global solutions. On the basis of networks consisting of one-dimensional domains, we present an extension of the RBEM which remedies this problem and provides a good basis representation for each individual edge. Essentially this extension makes use of a splinebased boundary parametrization in the local basis construction. To substantiate the approximation properties of the basis representation onto the global solution, we develop an error estimate for local basis construction with Proper Orthogonal Decomposition (POD) or POD-Greedy. Additionally, we provide existence, uniqueness and regularity results for parabolic PDEs on networks with one-dimensional domains, which are essential for the error analysis. Finally, we illustrate our method with three examples. The first corresponds to the theory presented and shows two different networks of one-dimensional heat equations with varying thermal conductivity. The second and third problem demonstrates the extensibility of the method to component based domains in two dimensions or nonlinear PDEs. These were parts of the research project Life-cycle oriented optimization for a resource and energy efficient infrastructure, funded by the German Federal Ministry of Education and Research. |
| format | Online |
| id | doab-20.500.12854ir-170208 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | FAU University Press |
| publisherStr | FAU University Press |
| record_format | ojs |
| spelling | doab-20.500.12854ir-1702082025-12-16T05:20:24Z Simulation-Based Model Reduction for Partial Differential Equations on Networks Walther, Maximilian Greedy-Algorithmus Ordnungsreduktion Parabolische Differentialgleichung Fehlerrechnung POD-Methode thema EDItEUR::P Mathematics and Science In this thesis, we consider model reduction for parameter dependent parabolic PDEs defined on networks with variable composition. For this type of problem, the Reduced Basis Element Method (RBEM), developed by Maday and Rønquist, is a reasonable choice as a solution on the entire domain is not required. The reduction method is based on the idea of constructing a reduced basis for every individual component and coupling the reduced elements using a mortar-like method. However, this decomposition procedure can lead to difficulties, especially for networks consisting of numerous edges. Due to the variable composition of the networks, the solution on the interfaces is extremely difficult to predict. This can lead to unsuitable basis functions and poor approximations of the global solutions. On the basis of networks consisting of one-dimensional domains, we present an extension of the RBEM which remedies this problem and provides a good basis representation for each individual edge. Essentially this extension makes use of a splinebased boundary parametrization in the local basis construction. To substantiate the approximation properties of the basis representation onto the global solution, we develop an error estimate for local basis construction with Proper Orthogonal Decomposition (POD) or POD-Greedy. Additionally, we provide existence, uniqueness and regularity results for parabolic PDEs on networks with one-dimensional domains, which are essential for the error analysis. Finally, we illustrate our method with three examples. The first corresponds to the theory presented and shows two different networks of one-dimensional heat equations with varying thermal conductivity. The second and third problem demonstrates the extensibility of the method to component based domains in two dimensions or nonlinear PDEs. These were parts of the research project Life-cycle oriented optimization for a resource and energy efficient infrastructure, funded by the German Federal Ministry of Education and Research. 2025-12-16T05:20:23Z 2025-12-16T05:20:23Z 2025-12-15T15:09:48Z 2018 book ONIX_20251215T160703_9783961471560_24 https://library.oapen.org/handle/20.500.12657/109193 9783961471560 9783961471553 https://directory.doabooks.org/handle/20.500.12854/170208 eng FAU Studies Mathematics & Physics open access image/jpeg Attribution-NonCommercial-NoDerivatives 4.0 International https://library.oapen.org/bitstream/20.500.12657/109193/1/9783961471560.pdf FAU University Press 10.25593/978-3-96147-156-0 10.25593/978-3-96147-156-0 2c600dea-eece-4066-87be-da335e323fdb 9783961471560 9783961471553 183 Erlangen open access |
| spellingShingle | Greedy-Algorithmus Ordnungsreduktion Parabolische Differentialgleichung Fehlerrechnung POD-Methode thema EDItEUR::P Mathematics and Science Walther, Maximilian Simulation-Based Model Reduction for Partial Differential Equations on Networks |
| title | Simulation-Based Model Reduction for Partial Differential Equations on Networks |
| title_full | Simulation-Based Model Reduction for Partial Differential Equations on Networks |
| title_fullStr | Simulation-Based Model Reduction for Partial Differential Equations on Networks |
| title_full_unstemmed | Simulation-Based Model Reduction for Partial Differential Equations on Networks |
| title_short | Simulation-Based Model Reduction for Partial Differential Equations on Networks |
| title_sort | simulation based model reduction for partial differential equations on networks |
| topic | Greedy-Algorithmus Ordnungsreduktion Parabolische Differentialgleichung Fehlerrechnung POD-Methode thema EDItEUR::P Mathematics and Science |
| topic_facet | Greedy-Algorithmus Ordnungsreduktion Parabolische Differentialgleichung Fehlerrechnung POD-Methode thema EDItEUR::P Mathematics and Science |
| url | ONIX_20251215T160703_9783961471560_24 |
| work_keys_str_mv | AT walthermaximilian simulationbasedmodelreductionforpartialdifferentialequationsonnetworks |