An extension of the low-rank Lyapunov ADI to non-zero initial values and its applications
<|reference_start|>An extension of the low-rank Lyapunov ADI to non-zero initial values and its applications: We derive the Alternating-Direction Implicit (ADI) method based on a commuting operator split and apply the results to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution. Previously, it has been mandatory to start the low-rank ADI (LR-ADI) with an all-zero initial value. Our approach retains the known efficient iteration schemes of low-rank increments and residual to arbitrary low-rank initial values for the LR-ADI method. We further generalize some of the known properties of the LR-ADI for Lyapunov equations to larger classes of algorithms or problems.   We investigate the performance of arbitrary initial values using two outer iterations in which LR-ADI is typically called. First, we solve an algebraic Riccati equation with the Newton method. Second, we solve a differential Riccati equation with a first-order Rosenbrock method. Numerical experiments confirm that the proposed new initial value of the alternating-directions implicit (ADI) can lead to a significant reduction in the total number of ADI steps, while also showing a 17% and 8x speed-up over the zero initial value for the two equation types, respectively.<|reference_end|>
@article{schulze2024an,
    title={An extension of the low-rank Lyapunov ADI to non-zero initial values and
  its applications},
    author={Jonas Schulze, Jens Saak},
    journal={arXiv preprint arXiv:2406.13477},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.13477},
    primaryClass={math.NA cs.NA}
}