A projector-splitting integrator for dynamical low-rank approximation
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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.
KeywordsLow-rank approximation Time-dependent matrices Matrix differential equations Numerical integrator
Mathematics Subject Classification (2010)65F30 65L05 65L20 15A23
- 15.Oseledets, I.V., Khoromskij, B.N., Schneider, R.: Efficient time-stepping scheme for dynamics on TT-manifolds. Preprint 24, MPI MIS, 2012 Google Scholar