To enhance the performance of retraction-based symplectic integrators beyond the suggestions in the paper, several strategies can be considered:
Adaptive Time-Stepping: Implementing an adaptive time-stepping scheme can dynamically adjust the step size based on the local error estimate. This can improve accuracy without increasing computational cost unnecessarily.
Higher-Order Methods: Exploring higher-order symplectic integrators, such as higher-order Runge-Kutta methods or implicit methods like the implicit midpoint rule, can provide better accuracy and stability over longer integration periods.
Parallelization: Utilizing parallel computing techniques can distribute the computational load across multiple processors, speeding up the integration process and enabling the handling of more complex problems.
Optimized Algorithms: Fine-tuning the implementation of the retraction-based integrators, optimizing the code for efficiency, and leveraging hardware acceleration (e.g., GPU computing) can significantly improve performance.
Hybrid Methods: Combining retraction-based integrators with other numerical methods, such as spectral methods or machine learning techniques, can potentially enhance accuracy and efficiency in approximating Riemannian cubic polynomials on Stiefel manifolds.