线性矩阵方程ADI类方法的统一框架及其有益后果
A unifying framework for ADI-like methods for linear matrix equations and beneficial consequences
AI总结：
本文提出基于交换算子分裂的统一ADI框架，将低秩ADI方法扩展至任意初始值，并证明其性能优势，在两类Riccati方程求解中分别获得17%和8倍加速。
摘要：
我们基于交换算子分裂推导了交替方向隐式（ADI）方法，并将结果详细应用于具有低秩常数项和近似解的连续时间代数Lyapunov方程，同时为Sylvester方程情形提供了指引。此前，对于Lyapunov方程的低秩ADI（CF-ADI、LR-ADI、G-LR-ADI）或Sylvester方程（fADI、G-fADI），必须从全零初始值开始。我们的方法将已知的低秩增量和残差的高效迭代方案扩展到所有这些方法中任意低秩初始值的情形。我们进一步将低秩Lyapunov ADI的两个性质推广到应用于任意线性方程的通用ADI（基于交换算子分裂），即迭代在移位参数置换下的不变性，以及复数移位参数的高效处理。我们通过两种通常调用低秩Lyapunov ADI的外迭代来研究任意初始值的性能。首先，我们使用牛顿法求解代数Riccati方程。其次，我们使用一阶Rosenbrock方法求解微分Riccati方程。数值实验证实，所提出的ADI新初始值可以显著减少ADI步骤的总数，同时对于这两种方程类型，与零初始值相比分别实现了17%和8倍的加速。
英文摘要：
We derive the alternating-directions implicit (ADI) method based on a commuting operator split and apply the results in detail to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution, giving pointers for the Sylvester case. Previously, it has been mandatory to start the low-rank ADI for Lyapunov equations (CF-ADI, LR-ADI, G-LR-ADI) or Sylvester equations (fADI, G-fADI) with an all-zero initial value. Our approach extends the known efficient iteration schemes of low-rank increments and residuals to arbitrary low-rank initial values for all these methods. We further generalize two properties of the low-rank Lyapunov ADI to the generic ADI applied to arbitrary linear equations using a commuting operator split, namely the invariance of iterates under permutations of the shift parameters, and the efficient handling of complex shift parameters. We investigate the performance of arbitrary initial values using two outer iterations in which the low-rank Lyapunov 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 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.
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