Low-rank cross approximation of function-valued tensors for reduced-order modeling of parametric PDEs
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912734712430592 |
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| author | Budzinskiy, Stanislav Kazeev, Vladimir Olshanskii, Maxim |
| author_facet | Budzinskiy, Stanislav Kazeev, Vladimir Olshanskii, Maxim |
| contents | The paper considers function-valued tensors, viewed as multidimensional arrays with entries in an abstract Hilbert space. Despite the absence of the algebraic structure of a field, the geometric inner-product structure suffices to introduce the Tucker rank, higher-order SVD, and Tucker-cross decomposition for function-valued tensors. An adaptive cross-approximation algorithm is developed to compute low-rank approximations of such tensors. The framework is motivated by, and applied to, model order reduction of the parameter-to-solution map for a parametric PDE. The resulting reduced-order model can be interpreted as an encoder-decoder scheme with a nonlinear encoder and a multilinear decoder. The performance of the proposed non-intrusive approximation method is demonstrated in numerical examples for two nonlinear parametric PDE systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-rank cross approximation of function-valued tensors for reduced-order modeling of parametric PDEs Budzinskiy, Stanislav Kazeev, Vladimir Olshanskii, Maxim Numerical Analysis The paper considers function-valued tensors, viewed as multidimensional arrays with entries in an abstract Hilbert space. Despite the absence of the algebraic structure of a field, the geometric inner-product structure suffices to introduce the Tucker rank, higher-order SVD, and Tucker-cross decomposition for function-valued tensors. An adaptive cross-approximation algorithm is developed to compute low-rank approximations of such tensors. The framework is motivated by, and applied to, model order reduction of the parameter-to-solution map for a parametric PDE. The resulting reduced-order model can be interpreted as an encoder-decoder scheme with a nonlinear encoder and a multilinear decoder. The performance of the proposed non-intrusive approximation method is demonstrated in numerical examples for two nonlinear parametric PDE systems. |
| title | Low-rank cross approximation of function-valued tensors for reduced-order modeling of parametric PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.22650 |