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A rank-adaptive robust integrator for dynamical low-rank approximation

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Abstract

A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator.

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Acknowledgements

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 258734477—SFB 1173. It started out from a discussion at the 2021 annual meeting of SFB 1173 that included Marlis Hochbruck and Stefan Schrammer. We thank Martin Frank for fruitful discussions on radiation transport theory.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Gianluca Ceruti & Christian Lubich

  2. Karlsruhe Institute of Technology, Kaiserstr. 12, 76131, Karlsruhe, Germany

    Jonas Kusch

Authors
  1. Gianluca Ceruti
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  2. Jonas Kusch
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  3. Christian Lubich
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Correspondence to Gianluca Ceruti.

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Communicated by Antonella Zanna Munthe-Kaas.

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Ceruti, G., Kusch, J. & Lubich, C. A rank-adaptive robust integrator for dynamical low-rank approximation. Bit Numer Math 62, 1149–1174 (2022). https://doi.org/10.1007/s10543-021-00907-7

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