D-RADI：求解大规模离散时间代数Riccati方程的低秩ADI算法
D-RADI: A Low-rank ADI Algorithm for Solving Large-scale Discrete-time Algebraic Riccati Equations
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中文总结 AI 辅助
本文针对大规模离散时间代数Riccati方程（DARE），提出了一种自动生成ADI位移的低秩ADI求解器，经中等规模及10^6阶大规模问题验证，该求解器高效、精确且完全自主。
中文摘要 AI 辅助
低秩交替方向隐式（ADI）方法是求解多种可接受低秩解的大规模矩阵方程的高效数值技术。离散时间代数Riccati方程（DARE）是一类重要的矩阵方程，应用于状态估计、控制器设计和滤波器设计领域。文献中，针对Stein方程的低秩Cholesky因子ADI方法已被用于牛顿迭代中求解大规模DARE，但目前尚无针对此类DARE的专用低秩ADI求解器。为解决这一空白，本文提出了一种用于大规模DARE的低秩ADI求解器，还提出了一种自动生成ADI位移的高效方法，使所提求解器在求解DARE时完全自主。在中等规模问题上，将所提求解器与MATLAB的idare进行了比较；在阶数为10^6的大规模DARE上进一步验证了其效率和精度。数值结果证实该求解器高效、精确且完全自主。
英文摘要
The low-rank alternating direction implicit (ADI) method is an efficient numerical technique for solving several types of large-scale matrix equations that admit low-rank solutions. The discrete-time algebraic Riccati equation (DARE) is an important matrix equation with applications in state estimation, controller design, and filter design. In the literature, the low-rank Cholesky factor ADI method for Stein equations has been used within Newton iterations to solve large-scale DAREs. However, no dedicated low-rank ADI solver is available for such DAREs. To address this gap, this paper presents a low-rank ADI solver for large-scale DAREs. We also propose an efficient approach to generate ADI shifts automatically, which makes the proposed solver fully autonomous for solving DAREs. The effectiveness of the proposed solver is compared with MATLAB's \texttt{idare} on a moderate-order problem. Efficiency and accuracy are further demonstrated on large-scale DAREs of order $10^6$. Numerical results confirm that the solver is efficient, accurate, and fully autonomous.
发表机构
- Yangtze University（长江大学）
机构由 AI 辅助整理，请以论文原文为准。
