A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913818398949376 |
|---|---|
| author | Mueller, Nicholas Zhao, Yiran Badia, Santiago Cui, Tiangang |
| author_facet | Mueller, Nicholas Zhao, Yiran Badia, Santiago Cui, Tiangang |
| contents | In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids Mueller, Nicholas Zhao, Yiran Badia, Santiago Cui, Tiangang Numerical Analysis 65M60 G.1.2; G.1.8 In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques. |
| title | A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids |
| topic | Numerical Analysis 65M60 G.1.2; G.1.8 |
| url | https://arxiv.org/abs/2412.14460 |