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Projected exponential methods for stiff dynamical low-rank approximation problems

Benjamin Carrel, Bart Vandereycken

TL;DR

The paper develops projected exponential methods for stiff dynamical low-rank approximation problems, focusing on Sylvester-like matrix differential equations where storing full solutions is infeasible. It introduces two concrete schemes, projected exponential Euler (order 1) and projected exponential Runge (order 2), and proves their convergence under standard DLRA assumptions, with explicit dependence on one-sided Lipschitz constants and the rank-truncation error. A Krylov-based implementation framework (including rational and lucky Krylov strategies) enables efficient, memory-conscious evaluation of $\varphi$-functions while preserving low-rank structure and providing a priori error guidance. Numerical experiments on differential Lyapunov, Riccati, and Allen–Cahn equations demonstrate stiffness robustness, speedups over existing DLRA methods, and effective rank-adaptivity, supporting practical applicability to high-dimensional, stiff problems. Overall, the projected exponential methods offer a principled, scalable approach for accurate, low-memory integration of stiff, high-dimensional DLRA problems with rigorous error control.

Abstract

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low- \rank approximation is a recent technique for solving high-dimensional differential equations by means of low-rank approximations. However, the domain is lacking numerical methods for stiff equations since existing methods are either not robust-to-stiffness or have unreasonably large hidden constants. In this paper, we focus on solving large-scale stiff matrix differential equations with a Sylvester-like structure, that admit good low-rank approximations. We propose two new methods that have good convergence properties, small memory footprint and that are fast to compute. The theoretical analysis shows that the new methods have order one and two, respectively. We also propose a practical implementation based on Krylov techniques. The approximation error is analyzed, leading to a priori error bounds and, therefore, a mean for choosing the size of the Krylov space. Numerical experiments are performed on several examples, confirming the theory and showing good speedup in comparison to existing techniques.

Projected exponential methods for stiff dynamical low-rank approximation problems

TL;DR

The paper develops projected exponential methods for stiff dynamical low-rank approximation problems, focusing on Sylvester-like matrix differential equations where storing full solutions is infeasible. It introduces two concrete schemes, projected exponential Euler (order 1) and projected exponential Runge (order 2), and proves their convergence under standard DLRA assumptions, with explicit dependence on one-sided Lipschitz constants and the rank-truncation error. A Krylov-based implementation framework (including rational and lucky Krylov strategies) enables efficient, memory-conscious evaluation of -functions while preserving low-rank structure and providing a priori error guidance. Numerical experiments on differential Lyapunov, Riccati, and Allen–Cahn equations demonstrate stiffness robustness, speedups over existing DLRA methods, and effective rank-adaptivity, supporting practical applicability to high-dimensional, stiff problems. Overall, the projected exponential methods offer a principled, scalable approach for accurate, low-memory integration of stiff, high-dimensional DLRA problems with rigorous error control.

Abstract

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low- \rank approximation is a recent technique for solving high-dimensional differential equations by means of low-rank approximations. However, the domain is lacking numerical methods for stiff equations since existing methods are either not robust-to-stiffness or have unreasonably large hidden constants. In this paper, we focus on solving large-scale stiff matrix differential equations with a Sylvester-like structure, that admit good low-rank approximations. We propose two new methods that have good convergence properties, small memory footprint and that are fast to compute. The theoretical analysis shows that the new methods have order one and two, respectively. We also propose a practical implementation based on Krylov techniques. The approximation error is analyzed, leading to a priori error bounds and, therefore, a mean for choosing the size of the Krylov space. Numerical experiments are performed on several examples, confirming the theory and showing good speedup in comparison to existing techniques.
Paper Structure (20 sections, 7 theorems, 147 equations, 9 figures)

This paper contains 20 sections, 7 theorems, 147 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Exponential integrators
  4. Dynamical low-rank approximation
  5. Contributions and outline
  6. Projected exponential methods
  7. Preliminary results
  8. Projected exponential Euler
  9. Projected exponential Runge
  10. Efficient implementation with Krylov subspace techniques
  11. Rational Krylov approximation
  12. Lucky Krylov approximation
  13. A priori approximation bounds
  14. Numerical experiments
  15. Application to the differential Lyapunov equation
  16. ...and 5 more sections

Key Result

Theorem 4

Assume that $\mathcal{G}$ is locally Lipschitz-continuous in a strip along the exact solution. Let $X_n^{\mathrm{E}}$ be the $n$-th iteration of the exponential Euler scheme eq:exponential_euler, and let $X(t_n)$ be the solution to eq:full problem at time $t_n = nh$. Then, for all $h \leq h_0$, the where the constant $C$ is explicitly available in the proof and depends on $\ell, L_{\mathcal{G}},

Figures (9)

  • Figure 1: Error of several methods under mesh refinements for solving the heat equation (stiff) that can be formulated as differential Lyapunov equation with constant source $C$; see Section \ref{['sec:diff Lyap']}. Step size is $h = 0.01$.
  • Figure 2: New methods applied to the differential Lyapunov equation with time-dependent source. The low-rank splitting techniques are from ostermann2019convergence.
  • Figure 3: Projected exponential Runge \ref{['eq: proj exp Runge']} applied to differential Lyapunov equation with time-dependent source. The dashed lines indicate the minimal error for that particular rank.
  • Figure 4: New methods applied to the differential Lyapunov equation with time-dependent source.
  • Figure 5: The new methods applied to the differential Riccati equation. The low-rank splitting methods are from ostermann2019convergence.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 1: One-sided Lipschitz
  • Definition 2: Lipschitz continuity
  • Remark 1
  • Definition 3: $\varphi$-functions skaflestad2009scaling
  • Definition 4: Logarithmic norm
  • proof
  • Theorem 4: Convergence of exponential Euler
  • proof
  • Theorem 6: Convergence of exponential Runge
  • proof
  • ...and 17 more