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Randomized low-rank Runge-Kutta methods

Hei Yin Lam, Gianluca Ceruti, Daniel Kressner

Abstract

This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank approximation to keep the rank of the stages limited. The so-called generalized Nyström method is particularly well suited for this purpose; it builds low-rank approximations from random sketches of the discretized dynamics. In contrast, all existing dynamical low-rank approximation methods are deterministic and usually perform tangent space projections to limit rank growth. Using such tangential projections can result in larger error compared to approximating the dynamics directly. Moreover, sketching allows for increased flexibility and efficiency by choosing structured random matrices adapted to the structure of the matrix differential equation. Under suitable assumptions, we establish moment and tail bounds on the error of our randomized low-rank Runge-Kutta methods. When combining the classical Runge-Kutta method with generalized Nyström, we obtain a method called Rand RK4, which exhibits fourth-order convergence numerically -- up to the low-rank approximation error. For a modified variant of Rand RK4, we also establish fourth-order convergence theoretically. Numerical experiments for a range of examples from the literature demonstrate that randomized low-rank Runge-Kutta methods compare favorably with two popular dynamical low-rank approximation methods, in terms of robustness and speed of convergence.

Randomized low-rank Runge-Kutta methods

Abstract

This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank approximation to keep the rank of the stages limited. The so-called generalized Nyström method is particularly well suited for this purpose; it builds low-rank approximations from random sketches of the discretized dynamics. In contrast, all existing dynamical low-rank approximation methods are deterministic and usually perform tangent space projections to limit rank growth. Using such tangential projections can result in larger error compared to approximating the dynamics directly. Moreover, sketching allows for increased flexibility and efficiency by choosing structured random matrices adapted to the structure of the matrix differential equation. Under suitable assumptions, we establish moment and tail bounds on the error of our randomized low-rank Runge-Kutta methods. When combining the classical Runge-Kutta method with generalized Nyström, we obtain a method called Rand RK4, which exhibits fourth-order convergence numerically -- up to the low-rank approximation error. For a modified variant of Rand RK4, we also establish fourth-order convergence theoretically. Numerical experiments for a range of examples from the literature demonstrate that randomized low-rank Runge-Kutta methods compare favorably with two popular dynamical low-rank approximation methods, in terms of robustness and speed of convergence.
Paper Structure (20 sections, 9 theorems, 89 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 9 theorems, 89 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and an idealized randomized projection method
  3. Assumptions
  4. Low-rank approximability
  5. Randomized low-rank approximation
  6. Idealized randomized projection method
  7. Error analysis
  8. Randomized low-rank Runge-Kutta methods
  9. Implementation aspects and cost
  10. Comparison to Projected Runge-Kutta method
  11. Error analysis
  12. Error analysis of randomized low-rank Runge-Kutta 4 method
  13. Numerical Experiments
  14. Lyapunov matrix differential equation
  15. Speeding up by constant random matrices
  16. ...and 5 more sections

Key Result

Theorem 2

Suppose that $\Omega\in \mathbb{R}^{n\times (r+p)}$ and $\Psi\in \mathbb{R}^{m\times (r+p+\ell)}$ are independent standard Gaussian matrices with $p,\ell\geq 4$. Setting $q=\min\{p,\ell\}$, it holds for $Z\in \mathbb{R}^{m\times n}$ that with $C_{\mathcal{N}}=1+2\sqrt{(1+r+p)(1+r)}$.

Figures (7)

  • Figure 1: Lyapunov matrix differential equation with $\alpha=1$. The singular values of the reference solution at time $T = 1$ together with the approximation errors of the numerical approximation obtained via the Rand Euler and Rand RK4 for different ranks and time-step sizes.
  • Figure 2: Lyapunov matrix differential equation with $\alpha=10^{-5}$ and $\alpha=1$. Comparison of absolute approximation errors for different low-rank integrators using rank $r = 10$.
  • Figure 3: Lyapunov matrix differential equation with $\alpha=1$. Comparison of absolute approximation errors for Rand RK2, Rand RK2 with different (independent) random matrices across stages vs. Rand RK2 and Rand RK4 with the same random matrices across stages.
  • Figure 4: Non-linear Schrödinger equation with $\alpha=0.3$. Singular values of the reference solution at time $T = 5$ together with the approximation errors of the numerical approximation obtained by Rand Euler and Rand RK4 for different ranks and time-step sizes.
  • Figure 5: Non-linear Schrödinger equation with $\alpha=3\times 10^{-4}$ and $\alpha=3\times 10^{-1}$. Comparison of absolute approximation errors measured in Frobenius norm for different integrators for rank-30.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Example 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • proof
  • Remark 6
  • ...and 11 more