Combinatorics

Tensor (intrinsic definition)

Bandwidth (computing)

Projector

Computer network

Mathematics

Integrator

Gene

Matrix (chemical analysis)

Rank (graph theory)

Mathematical analysis

Artificial intelligence

Composite material

Control (management)

Chemistry

Materials science

Geometry

Computer science

Applied mathematics

Computer vision

Control theory (sociology)

Biochemistry

Robustness (evolution)

Low-rank approximation

BIT Numerical Mathematics

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.

An unconventional robust integrator for dynamical low-rank approximation