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

TL;DR

This paper introduces a rank-adaptive integrator for dynamical low-rank approximation that dynamically adjusts the rank during integration, maintaining key properties and improving flexibility in solving matrix and tensor differential equations.

Contribution

It extends a fixed-rank integrator to an adaptive version that automatically chooses rank, preserving important properties and broadening applicability.

Findings

Retains exactness, robustness, and symmetry-preserving properties.

Preserves norm, energy, and monotonic functional decrease under certain conditions.

Numerical experiments demonstrate effective adaptive behavior.

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 Schr\"odinger 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.