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Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations

Zi Wu, Yong-Liang Zhao, Xian-Ming Gu

TL;DR

This work tackles efficient computation of low-rank solutions to large-scale semilinear stiff matrix differential equations by integrating operator splitting with randomized dynamical low-rank approximations. It splits the dynamics into a stiff linear part solved exactly by an exponential integrator and a nonstiff nonlinear part approximated with dynamic rangefinding, DRSVD, or DGN within a rank-$r$ manifold, enabling Lie-Trotter and Strang schemes and rank adaptation. The main contributions are the development of a unified RDLR framework with three randomized nonlinear solvers, an explicit rank-adaptive extension, and comprehensive validation on canonical stiff problems such as the Allen-Cahn equation and differential Riccati equations, showing first-to-second-order temporal convergence and strong robustness. The results indicate a scalable, stable approach for high-dimensional PDE discretizations and control-theoretic problems, offering practical impact for simulations demanding large-scale matrix dynamics with stiffness and nonlinearity.

Abstract

In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose splitting-based randomized dynamical low-rank approximations for a low-rank solution of the stiff matrix differential equation. We first split such the equation into a stiff linear subproblem and a nonstiff nonlinear subproblem. Then, a low-rank exponential integrator is applied to the linear subproblem. Two randomized low-rank approaches are employed for the nonlinear subproblem. Furthermore, we extend the proposed methods to rank-adaptation scenarios. Through rigorous validation on canonical stiff matrix differential problems, including spatially discretized Allen-Cahn equations and differential Riccati equations, we demonstrate that our methods achieve desired convergence orders. Numerical results confirm the robustness and accuracy of the proposed methods.

Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations

TL;DR

This work tackles efficient computation of low-rank solutions to large-scale semilinear stiff matrix differential equations by integrating operator splitting with randomized dynamical low-rank approximations. It splits the dynamics into a stiff linear part solved exactly by an exponential integrator and a nonstiff nonlinear part approximated with dynamic rangefinding, DRSVD, or DGN within a rank- manifold, enabling Lie-Trotter and Strang schemes and rank adaptation. The main contributions are the development of a unified RDLR framework with three randomized nonlinear solvers, an explicit rank-adaptive extension, and comprehensive validation on canonical stiff problems such as the Allen-Cahn equation and differential Riccati equations, showing first-to-second-order temporal convergence and strong robustness. The results indicate a scalable, stable approach for high-dimensional PDE discretizations and control-theoretic problems, offering practical impact for simulations demanding large-scale matrix dynamics with stiffness and nonlinearity.

Abstract

In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose splitting-based randomized dynamical low-rank approximations for a low-rank solution of the stiff matrix differential equation. We first split such the equation into a stiff linear subproblem and a nonstiff nonlinear subproblem. Then, a low-rank exponential integrator is applied to the linear subproblem. Two randomized low-rank approaches are employed for the nonlinear subproblem. Furthermore, we extend the proposed methods to rank-adaptation scenarios. Through rigorous validation on canonical stiff matrix differential problems, including spatially discretized Allen-Cahn equations and differential Riccati equations, we demonstrate that our methods achieve desired convergence orders. Numerical results confirm the robustness and accuracy of the proposed methods.

Paper Structure

This paper contains 17 sections, 1 theorem, 55 equations, 8 figures, 9 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Splitting-based randomized dynamical low-rank approximations
  3. Splitting into two subproblems
  4. The dynamical low-rank integration
  5. The randomized dynamical low-rank methods
  6. The dynamical rangefinder
  7. The dynamical randomized SVD
  8. The dynamical generalised Nyström method
  9. Implementation and extension
  10. Implementation
  11. Extend to a rank-adaptive version
  12. Numerical results
  13. Convergence tests
  14. Allen-Cahn equation
  15. Differential Riccati equation
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

(woolfe2008fast) Let $A\in\mathbb{R}^{m\times n}$, fix a positive integer $\kappa$ and a real number $\alpha>1$. Draw an independent family $\{\omega^{(i)}:i=1,2,\ldots,\kappa\}$ of standard Gaussian vectors. Then, with probability $1-\alpha^{-\kappa}$.

Figures (8)

  • Figure 1: Comparison of relative errors between DRSVD, projector-splitting and DGN with different time step size $\tau$ for Eq. \ref{['eq4.1']} for target rank $r = 16$ and the number of time steps $\hat{M} = 1024$. Left: The Lie-Trotter splitting is employed for decoupling the equation. Right: The Strang splitting is employed for decoupling the equation.
  • Figure 2: Comparison of relative errors between DRSVD, projector-splitting and DGN over time for Eq. \ref{['eq4.1']} for target rank $r = 16$ and the number of time steps $\hat{M} = 1024$. Left: The Lie-Trotter splitting is employed for decoupling the equation. Right: The Strang splitting is employed for decoupling the equation.
  • Figure 3: Comparison of relative errors between Lie-Trotter splitting, Strang splitting and optimal rank approximation over time for Eq. \ref{['eq4.1']} for target rank $r = 14$ and the number of time steps $\hat{M} = 128$. Left: The DRSVD method is employed for the nonlinear terms. Right: The DGN method is employed for the nonlinear terms.
  • Figure 4: Numerical rank evolution of Eq. \ref{['eq4.1']} for splitting methods combined with the adaptive low-rank solvers, where $(\hat{M},\hat{N})=(512,1024)$. Left: ADRSVD-LT; Right: ADRSVD-ST.
  • Figure 5: Comparison of relative errors between Lie-Trotter splitting, Strang splitting and best rank approximation over time for Eq. \ref{['eq4.2']} for target rank $r = 8$ and the number of time steps $\hat{M} = 512$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • lemma thmcounterlemma