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Second-order robust parallel integrators for dynamical low-rank approximation

Jonas Kusch

TL;DR

DLRA reduces memory and computation by representing large matrix solutions on a low-rank manifold, but curvature can limit time-step stability. This work extends the parallel BUG integrator to second order via careful basis augmentation, maintaining parallel updates with maximal rank $2r$ and yielding a rigorous second-order robust error bound that is independent of manifold curvature. Norm preservation is shown to hold up to truncation tolerance and $O(h^4)$ terms, aligning with structure-preserving goals. Numerical experiments on the discrete Schrödinger equation and a lattice radiative-transfer model confirm the theoretical improvements and demonstrate substantial speedups over higher-rank, sequential methods while achieving comparable accuracy.

Abstract

Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator's accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments.

Second-order robust parallel integrators for dynamical low-rank approximation

TL;DR

DLRA reduces memory and computation by representing large matrix solutions on a low-rank manifold, but curvature can limit time-step stability. This work extends the parallel BUG integrator to second order via careful basis augmentation, maintaining parallel updates with maximal rank and yielding a rigorous second-order robust error bound that is independent of manifold curvature. Norm preservation is shown to hold up to truncation tolerance and terms, aligning with structure-preserving goals. Numerical experiments on the discrete Schrödinger equation and a lattice radiative-transfer model confirm the theoretical improvements and demonstrate substantial speedups over higher-rank, sequential methods while achieving comparable accuracy.

Abstract

Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator's accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments.
Paper Structure (14 sections, 7 theorems, 68 equations, 5 figures)

This paper contains 14 sections, 7 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Dynamical low-rank approximation
  4. The parallel basis-update & Galerkin integrator
  5. Second-order parallel integrators
  6. Variant 1
  7. Computational costs
  8. Variant 2
  9. Robust error bound
  10. Norm preservation up to high order terms
  11. Numerical results
  12. Discrete Schrödinger equation
  13. Lattice testcase of radiative transfer
  14. Conclusion and outlook

Key Result

Theorem 2.2

Under the above Assumptions as:assumptions, the error of the parallel rank-adaptive integrator is bounded by where constants only depend on the Lipschitz constant and bound of ${\mathbf F}$, a bound of second derivatives of the exact solution, and an upper bound of the time stepsize.

Figures (5)

  • Figure 1: Comparison of relative approximation errors measured in Frobenius norm for different integrators for ranks $r\in\{5,10,15\}$. Solid lines indicate convergence slopes of orders one, two, and three.
  • Figure 2: Comparison of Frobenius norm error for different integrators for ranks $r\in\{5,10,15\}$. Solid lines indicate convergence slopes of orders one, two, three, and four.
  • Figure 3: Left: Spatial setup of the lattice. Right: Reference solution at time $T=3.2$ in logarithmic scale computed with the full-rank spherical harmonics (P$_N$) method using spherical harmonics up to degree $N = 21$.
  • Figure 4: Left: Scalar flux of the midpoint BUG integrator. Left: Scalar flux of version 2 of the second-order parallel BUG integrator. Both solutions use a fixed rank $r=10$ and are plotted at time $T=3.2$.
  • Figure 5: Left: Scalar flux of the midpoint BUG integrator. Left: Scalar flux of version 2 of the second-order parallel BUG integrator. Both solutions use a fixed rank $r=20$ and are plotted at time $T=3.2$.

Theorems & Definitions (15)

  • Theorem 2.2: CeKL23, Theorem 4.1
  • Theorem 4.1
  • Lemma 4.2
  • Proof 1
  • Lemma 4.3
  • Proof 2
  • Lemma 4.4
  • Proof 3
  • Theorem 4.5
  • Proof 4
  • ...and 5 more