Optimal morphings for model-order reduction for poorly reducible problems with geometric variability
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914120105721856 |
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| author | Kabalan, Abbas Casenave, Fabien Bordeu, Felipe Ehrlacher, Virginie Ern, Alexandre |
| author_facet | Kabalan, Abbas Casenave, Fabien Bordeu, Felipe Ehrlacher, Virginie Ern, Alexandre |
| contents | We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov $N$-width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first $r$ modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal morphings for model-order reduction for poorly reducible problems with geometric variability Kabalan, Abbas Casenave, Fabien Bordeu, Felipe Ehrlacher, Virginie Ern, Alexandre Numerical Analysis We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov $N$-width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first $r$ modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries. |
| title | Optimal morphings for model-order reduction for poorly reducible problems with geometric variability |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2502.11632 |