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A robust second-order low-rank BUG integrator based on the midpoint rule

Gianluca Ceruti, Lukas Einkemmer, Jonas Kusch, Christian Lubich

TL;DR

This work advances dynamical low-rank approximation by delivering a robust second-order BUG integrator based on the midpoint rule. It proves a local and global error analysis under Lipschitz-bounded vector fields with small normal components, yielding a bound $\| {\mathbf Y}_n - {\mathbf A}(t_n) \| \le c_0\delta + c_1 \hat{\varepsilon} + c_2 h\varepsilon + c_3 h^2 + c_4 n\vartheta$, and a local bound $\| \overline{\mathbf Y}_1 - {\mathbf A}(t_1) \| \le C(h^2 + h\varepsilon_r + \varepsilon_{\hat r})$. Numerical experiments on the heat equation, a non-stiff discrete Schrödinger equation, and the Vlasov-Poisson equation demonstrate robust second-order convergence for the Midpoint BUG variants, with energy and norm conservation, and show favorable performance relative to augmented BUG and projected midpoint methods. The results indicate improved stability and accuracy for large-scale, low-rank matrix differential equations, with practical implications for high-dimensional, structure-preserving simulations.

Abstract

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

A robust second-order low-rank BUG integrator based on the midpoint rule

TL;DR

This work advances dynamical low-rank approximation by delivering a robust second-order BUG integrator based on the midpoint rule. It proves a local and global error analysis under Lipschitz-bounded vector fields with small normal components, yielding a bound , and a local bound . Numerical experiments on the heat equation, a non-stiff discrete Schrödinger equation, and the Vlasov-Poisson equation demonstrate robust second-order convergence for the Midpoint BUG variants, with energy and norm conservation, and show favorable performance relative to augmented BUG and projected midpoint methods. The results indicate improved stability and accuracy for large-scale, low-rank matrix differential equations, with practical implications for high-dimensional, structure-preserving simulations.

Abstract

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.
Paper Structure (8 sections, 2 theorems, 42 equations, 5 figures)

This paper contains 8 sections, 2 theorems, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Recap: the augmented BUG integrator of CeKL22
  3. A midpoint BUG integrator
  4. Robust second-order error bound
  5. Numerical experiments
  6. Heat equation
  7. Non-stiff numerical test case: a discrete Schrödinger equation
  8. The Vlasov-Poisson equation

Key Result

theorem 1

Assume ${\mathbf A}(t_0)={\mathbf Y}_0={\mathbf U}_0{\mathbf S}_0{\mathbf V}_0^\top$ is of rank $r$. Then, the local error is bounded by where $C$ depends only on $L$ and $B$ in LB, on the bound of third derivatives of the exact solution ${\mathbf A}(t)$ of ode-mat, and on an upper bound of the stepsize $h$.

Figures (5)

  • Figure 1: First eight singular values of the reference solution at time $T=1$ together with the approximation errors of the numerical approximation obtained via the augmented BUG and the Midpoint BUG $3r$ and $4r$ variants for different ranks and time-step sizes.
  • Figure 2: Comparison for the non-stiff test case of the relative approximation errors measured in Frobenius norm among the projected low rank midpoint scheme following KiV19 and the different BUG integrators for various ranks and time-step sizes with final time $T=1$.
  • Figure 3: Comparison for the non-stiff test case of the relative approximation errors measured in Frobenius norm among the projected low rank midpoint scheme following KiV19 and the different BUG integrators for various ranks and time-step sizes with final time $T=10$.
  • Figure 4: Comparison of the absolute error in norm and energy conservation up to $T=10$ using a time step of $h=0.05$, for the rank-20 numerical approximation obtained with various numerical integrators. The projected midpoint low-rank (MPLR) method is indicated by a dashed line, while the augmented BUG and midpoint BUGs are depicted with solid lines.
  • Figure 5: Comparison of the relative error for the Vlasov--Poisson equation among the projector-splitting integrators and the various BUG numerical integrators up to final time $T=10$.

Theorems & Definitions (8)

  • remark thmcounterremark: Variants
  • remark thmcounterremark: Trapezoidal rule
  • remark thmcounterremark: Structure preservation
  • theorem 1: Local error bound
  • remark thmcounterremark: Rank truncation
  • remark thmcounterremark: Variants
  • proof
  • theorem 2: Robust second-order global error bound