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A low-rank algorithm for solving Lyapunov operator $\varphi$-functions within the matrix-valued exponential integrators

Dongping Li, Xiuying Zhang, Hongjiong Tian

TL;DR

The paper tackles the challenge of efficiently computing Lyapunov operator $\varphi$-functions needed by matrix-valued exponential integrators for large-scale matrix differential equations, where low-rank structure is common. It introduces a scalable, low-rank algorithm built on a scaling and recursive procedure, backed by a quasi-backward error analysis to select parameters, and an LDL^T-based decomposition to maintain low rank throughout computations. Key contributions include a detailed parameter-selection framework for $m$ and $s$, a practical low-rank implementation $\texttt{phi\_lyap\_ldl}$, and numerical demonstrations showing substantial speedups over dense or Krylov-based approaches while preserving high accuracy for DLEs and DREs. The approach enables robust, scalable matrix-valued exponential integration in large-scale settings and can be extended to Sylvester operators and adaptive schemes, with future work on preprocessing techniques to further improve performance.

Abstract

In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator $\varphi$-functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank.The proposed method employs a scaling and recursive procedure, complemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost is primarily determined by the multiplication of sparse matrices with block vectors. Numerical experiments validate the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in solving differential Lyapunov equations and Riccati equations.

A low-rank algorithm for solving Lyapunov operator $\varphi$-functions within the matrix-valued exponential integrators

TL;DR

The paper tackles the challenge of efficiently computing Lyapunov operator -functions needed by matrix-valued exponential integrators for large-scale matrix differential equations, where low-rank structure is common. It introduces a scalable, low-rank algorithm built on a scaling and recursive procedure, backed by a quasi-backward error analysis to select parameters, and an LDL^T-based decomposition to maintain low rank throughout computations. Key contributions include a detailed parameter-selection framework for and , a practical low-rank implementation , and numerical demonstrations showing substantial speedups over dense or Krylov-based approaches while preserving high accuracy for DLEs and DREs. The approach enables robust, scalable matrix-valued exponential integration in large-scale settings and can be extended to Sylvester operators and adaptive schemes, with future work on preprocessing techniques to further improve performance.

Abstract

In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator -functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank.The proposed method employs a scaling and recursive procedure, complemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost is primarily determined by the multiplication of sparse matrices with block vectors. Numerical experiments validate the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in solving differential Lyapunov equations and Riccati equations.
Paper Structure (7 sections, 2 theorems, 63 equations, 5 figures, 2 tables, 5 algorithms)

This paper contains 7 sections, 2 theorems, 63 equations, 5 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Full-rank computing
  3. The scaling and recursive procedure
  4. Choice of the parameters $m$ and $s$
  5. low-rank implementation
  6. Numerical experiments
  7. Conclusion

Key Result

Theorem 1

Assume that $s^{-1}\mathcal{L}_A\in \Omega_m$ and the inverse of $\mathcal{L}_A$ exists. Then the approximation $\widehat{\Phi}_s$ generated by Algorithm alg2.1 satisfies where

Figures (5)

  • Figure 1: Results of mExpeul at $t=1$ (left) and $t=5$ (right) for Example \ref{['exa1a']}, respectively.
  • Figure 2: Relative errors of exprb2, exprb3, and Ros2 under the discrete $L^2(0,0.1,\mathbb{R}^{N\times N})$-norm versus various time step sizes for the integration of each equation in Example \ref{['exa2']}.
  • Figure 3: Relative errors of exprb2, exprb3, and Ros2 under the discrete $L^2(0,0.1,\mathbb{R}^{N\times N})$-norm versus computation times for the integration of each equation in Example \ref{['exa2']}.
  • Figure 4: Relative errors of exprb2, exprb3, and Ros2 at $T=0.1$ versus various time step sizes for the integration of each equation in Example \ref{['exa2']}.
  • Figure 5: Relative errors of exprb2, exprb3, and Ros2 at $T=0.1$ versus computation times for the integration of each equation in Example \ref{['exa2']}.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Lemma 1
  • proof