Journal
BIT NUMERICAL MATHEMATICS
Volume 62, Issue 1, Pages 23-44Publisher
SPRINGER
DOI: 10.1007/s10543-021-00873-0
Keywords
Dynamical low-rank approximation; Structure-preserving integrator; Matrix and tensor differential equations; Tucker tensor format
Funding
- Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [258734477-SFB 1173]
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The study introduces a numerical integrator that computes low-rank approximations to large time-dependent matrices or tensors, offering robustness to small singular values and potential advantages over existing integrators. Numerical experiments demonstrate the performance of the proposed integrator.
We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.
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