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Adaptive low-rank exponential integrators for large-scale differential Riccati equation

Jinyi Li, Dongping Li, Hua Yang

Abstract

Matrix differential Riccati equation (DRE) typically exhibits transient and steady-state phases, posing challenges for fixed-step time integration methods, which may lack accuracy during transients or oversample in steady regimes. In this work, we propose adaptive low-rank matrix-valued exponential integrators for large-scale stiff DRE. The methods combine embedded exponential Rosenbrock-type schemes and adaptive step-size control, enabling an automatic adjustment to the evolving solution dynamics. This improves the accuracy during rapid transient phases while maintaining high accuracy in the steady state. Numerical experiments on benchmark problems demonstrate that the proposed adaptive integrators consistently improve accuracy and computational efficiency compared with fixed-step low-rank schemes.

Adaptive low-rank exponential integrators for large-scale differential Riccati equation

Abstract

Matrix differential Riccati equation (DRE) typically exhibits transient and steady-state phases, posing challenges for fixed-step time integration methods, which may lack accuracy during transients or oversample in steady regimes. In this work, we propose adaptive low-rank matrix-valued exponential integrators for large-scale stiff DRE. The methods combine embedded exponential Rosenbrock-type schemes and adaptive step-size control, enabling an automatic adjustment to the evolving solution dynamics. This improves the accuracy during rapid transient phases while maintaining high accuracy in the steady state. Numerical experiments on benchmark problems demonstrate that the proposed adaptive integrators consistently improve accuracy and computational efficiency compared with fixed-step low-rank schemes.

Paper Structure

This paper contains 9 sections, 45 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Constant-step low-rank exponential Rosenbrock integrators for DRE
  3. Matrix-valued exponential Rosenbrock integrators for DRE
  4. Low-rank implementation of matrix-valued exponential Rosenbrock integrators
  5. Adaptive low-rank exponential Rosenbrock integrators for DRE
  6. Embedded exponential Rosenbrock integrators pair
  7. Step-size selection
  8. Numerical experiments
  9. Conclusion

Figures (3)

  • Figure 1: Comparison between the adaptive and non-adaptive low-rank exponential integrators when integrating DRE over [0, 0.1] in Experiment \ref{['exam1']}. Left: Relative errors as a function of number of steps. Right: Relative errors as computational time.
  • Figure 2: Evolution of the reference, adaptive, and non-adaptive low-rank exponential integrators for DRE in Experiment \ref{['exam1']}. Left: Results over [0, 0.1]. Right: Zoom of the left panel.
  • Figure 3: Adaptive time step sizes and relative errors for varying tolerances for the DRE in Experiment \ref{['exam1']}. Left: Semi-log plot of adaptive time step sizes. Right: Relative errors at $t = 0.002$ as a function of prescribed tolerance.

Theorems & Definitions (2)

  • Example 1
  • Example 2