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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2303.10771 |
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| _version_ | 1866911850739793920 |
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| author | Nouy, Anthony Pasco, Alexandre |
| author_facet | Nouy, Anthony Pasco, Alexandre |
| contents | We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$, such as PBDW, yields a recovery error limited by the Kolmogorov width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to $\mathcal{M}$. In this paper, we propose a state estimation method relying on dictionary-based model reduction, where a space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of $\ell_1$-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_10771 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dictionary-based model reduction for state estimation Nouy, Anthony Pasco, Alexandre Numerical Analysis 65M32 (Primary) 62J07, 60B20 (Secondary) G.1.2; G.1.8 We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$, such as PBDW, yields a recovery error limited by the Kolmogorov width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to $\mathcal{M}$. In this paper, we propose a state estimation method relying on dictionary-based model reduction, where a space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of $\ell_1$-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees. |
| title | Dictionary-based model reduction for state estimation |
| topic | Numerical Analysis 65M32 (Primary) 62J07, 60B20 (Secondary) G.1.2; G.1.8 |
| url | https://arxiv.org/abs/2303.10771 |