Active Sampling of Interpolation Points to Identify Dominant Subspaces for Model Reduction
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910592700252160 |
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| author | Reddig, Celine Goyal, Pawan Duff, Igor Pontes Benner, Peter |
| author_facet | Reddig, Celine Goyal, Pawan Duff, Igor Pontes Benner, Peter |
| contents | Model reduction is an active research field to construct low-dimensional surrogate models of high fidelity to accelerate engineering design cycles. In this work, we investigate model reduction for linear structured systems using dominant reachable and observable subspaces. When the training set $-$ containing all possible interpolation points $-$ is large, then these subspaces can be determined by solving many large-scale linear systems. However, for high-fidelity models, this easily becomes computationally intractable. To circumvent this issue, in this work, we propose an active sampling strategy to sample only a few points from the given training set, which can allow us to estimate those subspaces accurately. To this end, we formulate the identification of the subspaces as the solution of the generalized Sylvester equations, guiding us to select the most relevant samples from the training set to achieve our goals. Consequently, we construct solutions of the matrix equations in low-rank forms, which encode subspace information. We extensively discuss computational aspects and efficient usage of the low-rank factors in the process of obtaining reduced-order models. We illustrate the proposed active sampling scheme to obtain reduced-order models via dominant reachable and observable subspaces and present its comparison with the method where all the points from the training set are taken into account. It is shown that the active sample strategy can provide us $17$x speed-up without sacrificing any noticeable accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03892 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Active Sampling of Interpolation Points to Identify Dominant Subspaces for Model Reduction Reddig, Celine Goyal, Pawan Duff, Igor Pontes Benner, Peter Machine Learning Numerical Analysis Dynamical Systems 15A24, 15A23, 34K06, 34K35, 93C05, 93C23, 41A05 Model reduction is an active research field to construct low-dimensional surrogate models of high fidelity to accelerate engineering design cycles. In this work, we investigate model reduction for linear structured systems using dominant reachable and observable subspaces. When the training set $-$ containing all possible interpolation points $-$ is large, then these subspaces can be determined by solving many large-scale linear systems. However, for high-fidelity models, this easily becomes computationally intractable. To circumvent this issue, in this work, we propose an active sampling strategy to sample only a few points from the given training set, which can allow us to estimate those subspaces accurately. To this end, we formulate the identification of the subspaces as the solution of the generalized Sylvester equations, guiding us to select the most relevant samples from the training set to achieve our goals. Consequently, we construct solutions of the matrix equations in low-rank forms, which encode subspace information. We extensively discuss computational aspects and efficient usage of the low-rank factors in the process of obtaining reduced-order models. We illustrate the proposed active sampling scheme to obtain reduced-order models via dominant reachable and observable subspaces and present its comparison with the method where all the points from the training set are taken into account. It is shown that the active sample strategy can provide us $17$x speed-up without sacrificing any noticeable accuracy. |
| title | Active Sampling of Interpolation Points to Identify Dominant Subspaces for Model Reduction |
| topic | Machine Learning Numerical Analysis Dynamical Systems 15A24, 15A23, 34K06, 34K35, 93C05, 93C23, 41A05 |
| url | https://arxiv.org/abs/2409.03892 |