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Mazen Ali ^{1} ^{,} ^{*} ,
Anthony Nouy ^{2}

27 April 2018

We are interested in applying SVD to more general spaces, the motivating example being the Sobolev space $H^1(\Omega)$ of weakly differentiable functions over a domain $\Omega\subset\R^d$. Controlling the truncation error in the energy norm is particularly interesting for PDE applications. To this end, one can apply SVD to tensor products in $H^1(\Omega_1)\otimes H^1(\Omega_2)$ with the induced tensor scalar product. However, the resulting space is not $H^1(\Omega_1\times\Omega_2)$ but is instead the space $H^1_{\text{mix}}(\Omega_1\times\Omega_2)$ of functions with mixed regularity. For large $d>2$ this poses a restrictive regularity requirement on $u\in H^1(\Omega)$. On the other hand, the space $H^1(\Omega)$ is not a tensor product Hilbert space, in particular $\|\cdot\|_{H^1}$ is not a reasonable crossnorm. Thus, we can not identify $H^1(\Omega)$ with the space of Hilbert Schmidt operators and apply SVD.

However, it is known that $H^1(\Omega)$ is isomorph (here written for $d=2$) to the Banach intersection space

\[H^1(\Omega_1\times\Omega_2)=H^1(\Omega_1)\otimes L_2(\Omega_2)\cap L_2(\Omega_1)\otimes
H^1(\Omega_2)\]

with equivalent norms. Each of the spaces in the intersection is a tensor product
Hilbert space where SVD applies.
We investigate several approaches to construct low-rank approximations for a function $u\in H^1(\Omega_1\times\Omega_2)$.

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ScienceOpen disciplines: | Applied mathematics, Applications, Statistics, Data analysis, Mathematics, Mathematical modeling & Computation |

Keywords: | low-rank, SVD, Sobolev Spaces |