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

Ceruti, Gianluca; Kusch, Jonas; Lubich, Christian

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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Volltext §
DOI: 10.5445/IR/1000131729
Veröffentlicht am 22.04.2021
Cover der Publikation
Zugehörige Institution(en) am KIT Sonderforschungsbereich 1173 (SFB 1173)
Institut für Angewandte und Numerische Mathematik (IANM)
Publikationstyp Forschungsbericht/Preprint
Publikationsdatum 03.04.2021
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000131729
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 18 S.
Serie CRC 1173 Preprint ; 2021/16
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter dynamical low-rank approximation, rank adaptivity, structure-preserving integrator, matrix and tensor differential equations
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