Time integration of rank-constrained Tucker tensors
For researchers in numerical linear algebra and tensor computations, this provides a robust integrator for time-dependent Tucker tensors, extending known matrix methods to higher-order tensors.
This paper extends the projector-splitting integrator for dynamical low-rank approximation from matrices to Tucker tensors, presenting a discrete time integration method that reconstructs rank-constrained Tucker tensors exactly and is robust to small singular values. Numerical examples from quantum dynamics and tensor optimization confirm the theoretical results.
Dynamical low-rank approximation in the Tucker tensor format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favorable properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with problems from quantum dynamics and tensor optimization methods illustrate our theoretical results.