On the approximation of vector-valued functions by volume sampling
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909280122175488 |
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| author | Kressner, Daniel Ni, Tingting Uschmajew, André |
| author_facet | Kressner, Daniel Ni, Tingting Uschmajew, André |
| contents | Given a Hilbert space $\mathcal H$ and a finite measure space $Ω$, the approximation of a vector-valued function $f: Ω\to \mathcal H$ by a $k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue-Bochner space $L^2(Ω;\mathcal H)$, the best possible subspace approximation error $d_k^{(2)}$ is characterized by the singular values of $f$. However, for practical reasons, $\mathcal U$ is often restricted to be spanned by point samples of $f$. We show that this restriction only has a mild impact on the attainable error; there always exist $k$ samples such that the resulting error is not larger than $\sqrt{k+1} \cdot d_k^{(2)}$. Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457-1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225-247, 2006) on column subset selection for matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03212 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the approximation of vector-valued functions by volume sampling Kressner, Daniel Ni, Tingting Uschmajew, André Numerical Analysis Functional Analysis Given a Hilbert space $\mathcal H$ and a finite measure space $Ω$, the approximation of a vector-valued function $f: Ω\to \mathcal H$ by a $k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue-Bochner space $L^2(Ω;\mathcal H)$, the best possible subspace approximation error $d_k^{(2)}$ is characterized by the singular values of $f$. However, for practical reasons, $\mathcal U$ is often restricted to be spanned by point samples of $f$. We show that this restriction only has a mild impact on the attainable error; there always exist $k$ samples such that the resulting error is not larger than $\sqrt{k+1} \cdot d_k^{(2)}$. Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457-1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225-247, 2006) on column subset selection for matrices. |
| title | On the approximation of vector-valued functions by volume sampling |
| topic | Numerical Analysis Functional Analysis |
| url | https://arxiv.org/abs/2304.03212 |