Low rank methods for a class of generalized Lyapunov equations and related issues
Low rank methods for a class of generalized Lyapunov equations and related issues
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DOI：
10.1007/s00211-013-0521-0
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发表时间：
2013-03
影响因子：
2.1
通讯作者：
P. Benner;T. Breiten
中科院分区：
文献类型：
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作者：
P. Benner;T. Breiten
本文研究了双线性随机控制中广义Lyapunov方程的可能的低秩解方法。我们证明，在某些假设下，我们可以预期解矩阵中的强奇异值衰减，从而允许低秩次近似。由于理论工具很强地利用了与标准线性Lyapunov方程的联系，我们甚至可以将结果推广到由张化线性方程组描述的二维情形。我们进一步给出了一些最常用的线性低阶解技术的合理推广，如交替方向隐式(ADI)迭代和Krylov-Plus-Inverted-Krylov(K-PIK)方法。通过一些用于双线性模型降阶领域的标准数值算例，我们将展示新方法的有效性。
In this paper, we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation, we can even extend the result to the-dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krylov-Plus-Inverted-Krylov (K-PIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction, we will show the efficiency of the new methods.