تقرير
On the approximation of vector-valued functions by volume sampling
| العنوان: | On the approximation of vector-valued functions by volume sampling |
|---|---|
| المؤلفون: | Kressner, Daniel, Ni, Tingting, Uschmajew, André |
| سنة النشر: | 2023 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Numerical Analysis, Mathematics - Functional Analysis |
| الوصف: | Given a Hilbert space $\mathcal H$ and a finite measure space $\Omega$, the approximation of a vector-valued function $f: \Omega \to \mathcal H$ 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 $L^2(\Omega;\mathcal H)$, 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. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2304.03212 |
| رقم الأكسشن: | edsarx.2304.03212 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |