An unconventional robust integrator for dynamical low-rank approximation

• Published in: BIT Vol. 62; no. 1; pp. 23 - 44
• Main Authors: Ceruti, Gianluca, Lubich, Christian
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2022; Springer Nature B.V
• Subjects: Antisymmetry; Approximation; Computational Mathematics and Numerical Analysis; Differential equations; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices (mathematics); Numeric Computing; Robustness (mathematics); Splitting; Symmetry; Tensors; Time dependence; Time integration; Dynamical low-rank approximation; 65L05; 65L70; 65L20; Tucker tensor format; 15A69; Structure-preserving integrator; Matrix and tensor differential equations
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-021-00873-0

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.