BIT Numerical Mathematics

, Volume 54, Issue 1, pp 171–188 | Cite as

A projector-splitting integrator for dynamical low-rank approximation

  • Christian LubichEmail author
  • Ivan V. Oseledets


The dynamical low-rank approximation of time-dependent matrices is a low-rank factorization updating technique. It leads to differential equations for factors of the matrices, which need to be solved numerically. We propose and analyze a fully explicit, computationally inexpensive integrator that is based on splitting the orthogonal projector onto the tangent space of the low-rank manifold. As is shown by theory and illustrated by numerical experiments, the integrator enjoys robustness properties that are not shared by any standard numerical integrator. This robustness can be exploited to change the rank adaptively. Another application is in optimization algorithms for low-rank matrices where truncation back to the given low rank can be done efficiently by applying a step of the integrator proposed here.


Low-rank approximation Time-dependent matrices Matrix differential equations Numerical integrator 

Mathematics Subject Classification (2010)

65F30 65L05 65L20 15A23 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Skolkovo Institute of Science and TechnologySkolkovoRussia
  3. 3.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

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