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On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problems

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      Abstract

      Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems and eigenvalue problems on Hilbert spaces. Although this question is central to the success of all existing solvers based on low-rank tensor techniques, very few of the results available so far allow to draw meaningful conclusions for higher dimensions. In this work, we develop a constructive framework to study low-rank approximability. One major assumption is that the involved linear operator admits a low-rank representation with respect to the chosen tensor format, a property that is known to hold in a number of applications. Simple conditions, which are shown to hold for a fairly general problem class, guarantee that our derived low-rank truncation error estimates do not deteriorate as the dimensionality increases.

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      Journal
      2014-06-26
      2016-01-07
      1406.7026

      http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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      math.NA quant-ph

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