A projector-splitting integrator for dynamical low-rank approximation

Abstract

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.

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Correspondence to Christian Lubich.

Additional information

This work was partially supported by DFG, SPP 1324, by RFBR grants 12-01-00546-a, 11-01-00549-a, 12-01-33013 mol-ved-a, RFBR-DFG grant 12-01-91333, by Federal program “Scientific and scientific-pedagogical personnel of innovative Russia” (contracts 16.740.12.0727, grants 8500 and 8235).

Communicated by Peter Benner.

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Lubich, C., Oseledets, I.V. A projector-splitting integrator for dynamical low-rank approximation. Bit Numer Math 54, 171–188 (2014). https://doi.org/10.1007/s10543-013-0454-0

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Keywords

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

Mathematics Subject Classification (2010)

  • 65F30
  • 65L05
  • 65L20
  • 15A23