2023/12/05 by Patrick Kürschner, Kürschner, Patrick
Computer Science · Engineering · Physics and Astronomy · #15A06 #15A24 #65F45 #65F55 #Adaptive optics and wavefront sensing #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Photonic and Optical Devices
paper · pdf · doi:10.48550/arxiv.2312.02891
openalex publication_date 2023/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the low-rank alternating directions implicit (ADI) iteration for approximately solving large-scale algebraic Sylvester equations. Inside every iteration step of this iterative process a pair of linear systems of equations has to be solved. We investigate the situation when those inner linear systems are solved inexactly by an iterative methods such as, for example, preconditioned Krylov subspace methods. The main contribution of this work are thresholds for the required accuracies regarding the inner linear systems which dictate when the employed inner Krylov subspace methods can be safely terminated. The goal is to save computational effort by solving the inner linear system as inaccurate as possible without endangering the functionality of the low-rank Sylvester-ADI method. Ideally, the inexact ADI method mimics the convergence behaviour of the more expensive exact ADI method, where the linear systems are solved directly. Alongside the theoretical results, also strategies for an actual practical implementation of the stopping criteria are developed. Numerical experiments confirm the effectiveness of the proposed strategies.