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

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      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
      Copyright © 2018

      This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

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