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Convergence of a Low-Rank Strang Splitting for Stiff Matrix Differential Equations

Carmen Scalone, Nicola Guglielmi

TL;DR

This work introduces a second-order Strang splitting method for stiff matrix differential equations of Sylvester-type, combining an exact exponential treatment of the linear stiff part with a dynamical low-rank approximation of the nonlinear part. The nonlinear flow is integrated using a second-order BUG2-based scheme, and low-rank preservation is enforced through tangent-space projections and rank truncations. The authors provide a rigorous convergence analysis showing global second-order accuracy under mild regularity and compatibility assumptions, with error decompositions that separate full-rank, initial-rank, and low-rank truncation effects. Numerical experiments on differential Lyapunov equations and semilinear parabolic problems confirm the theoretical results, demonstrate robustness to stiffness and rank truncation, and illustrate the method’s practical efficiency and adaptability in high-dimensional settings.

Abstract

We propose and analyze a second-order Strang splitting method for a class of stiff matrix differential equations with Sylvester-type structure. The method splits the dynamics into a stiff linear part, treated exactly via matrix exponentials, and a nonlinear part, integrated by a second-order dynamical low-rank (DLR) scheme. Our main contribution is a rigorous convergence proof showing that, under suitable assumptions, the overall scheme achieves second-order accuracy. Numerical experiments confirm the theoretical results and demonstrate the robustness and efficiency of the proposed method.

Convergence of a Low-Rank Strang Splitting for Stiff Matrix Differential Equations

TL;DR

This work introduces a second-order Strang splitting method for stiff matrix differential equations of Sylvester-type, combining an exact exponential treatment of the linear stiff part with a dynamical low-rank approximation of the nonlinear part. The nonlinear flow is integrated using a second-order BUG2-based scheme, and low-rank preservation is enforced through tangent-space projections and rank truncations. The authors provide a rigorous convergence analysis showing global second-order accuracy under mild regularity and compatibility assumptions, with error decompositions that separate full-rank, initial-rank, and low-rank truncation effects. Numerical experiments on differential Lyapunov equations and semilinear parabolic problems confirm the theoretical results, demonstrate robustness to stiffness and rank truncation, and illustrate the method’s practical efficiency and adaptability in high-dimensional settings.

Abstract

We propose and analyze a second-order Strang splitting method for a class of stiff matrix differential equations with Sylvester-type structure. The method splits the dynamics into a stiff linear part, treated exactly via matrix exponentials, and a nonlinear part, integrated by a second-order dynamical low-rank (DLR) scheme. Our main contribution is a rigorous convergence proof showing that, under suitable assumptions, the overall scheme achieves second-order accuracy. Numerical experiments confirm the theoretical results and demonstrate the robustness and efficiency of the proposed method.
Paper Structure (21 sections, 4 theorems, 58 equations, 5 figures, 1 table)

This paper contains 21 sections, 4 theorems, 58 equations, 5 figures, 1 table.

Table of Contents

  1. A low-rank approximation of stiff matrix differential equations
  2. Splitting into three subproblems
  3. The low-rank integrator
  4. The second order BUG integrator
  5. The augmented BUG integrator of CKL
  6. 1. Basis update (augmented bases).
  7. 2. Galerkin step.
  8. 3. Truncation.
  9. Midpoint BUG integrator (second order)
  10. Truncation:
  11. Full convergence analysis
  12. The main convergence result
  13. Detailed convergence proofs
  14. The error of the full-rank Strang splitting
  15. The low-rank Strang splitting
  16. ...and 6 more sections

Key Result

Theorem 1

Under Assumption hp_split, there exists $\tau_0 > 0$ such that, for all $0 < \tau \leq \tau_0$ and $n$ such that $t_0 + n\tau \leq T$, the global error satisfies where the constants $c_0, c_1, c_2$ depend on $\omega$, $C_s$, $L$, $B$ and $T$, but are independent of $\tau$, $\varepsilon$, $\delta$ and $n$. The constants $\delta$ and $\varepsilon$ are defined in delta and hp_splitass:MR.

Figures (5)

  • Figure 1: Error of the low-rank Strang splitting as a function of the time step $\tau$ and the approximation rank $r$, for problem \ref{['heat']}. Second-order convergence is observed when the rank is sufficiently high.
  • Figure 2: Error of the low-rank Strang splitting as a function of the time step and approximation rank for the Lyapunov equation with random low-rank forcing term. The convergence order deteriorates due to the lack of compatibility between the inhomogeneity and the Laplacian.
  • Figure 3: Decay of the first 10 singular values of the numerical reference solution, computed by the full-rank Strang splitting and time step $\tau = 10^{-6}$, at time $T=0.3$.
  • Figure 4: Frobenius norm error of the low-rank Strang splitting method as a function of time step and rank for problem \ref{['cubica']}. Second-order convergence is observed for ranks $\geq 4$.
  • Figure 5: Rank-adaptive integration for problem \ref{['cubica']}, starting from rank 3. Time step $\tau = 0.005$, truncation threshold $\theta = 10^{-8}$.

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1: Global error of the low-rank Strang splitting
  • Remark 2
  • Proposition 1: Global error of the full-rank Strang splitting
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['main_th']}
  • ...and 1 more