Adaptive low-rank exponential integrators for large-scale differential Riccati equation
This work addresses a domain-specific problem in numerical methods for control theory and large-scale systems, offering incremental improvements in adaptive time integration.
The authors tackled the challenge of efficiently solving large-scale stiff matrix differential Riccati equations, which have transient and steady-state phases, by proposing adaptive low-rank exponential integrators. The result was improved accuracy and computational efficiency compared to fixed-step methods, as demonstrated in numerical experiments.
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.