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# SVD on Intersection Spaces

, 1 , * , 2

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SVD, Sobolev Spaces, low-rank

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### Abstract

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 $u\in H^1(\Omega_1\times\Omega_2)$.

### Author and article information

###### Affiliations
[1 ]Ulm University
[2 ]Centrale Nantes, LMJL
[* ]Correspondence: mazen.ali@ 123456uni-ulm.de
###### Journal
ScienceOpen Posters
ScienceOpen
27 April 2018
10.14293/P2199-8442.1.SOP-MATH.WKIDDZ.v1