Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control
Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control
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DOI：
10.1007/s11075-015-9991-8
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发表时间：
2016-02
影响因子：
2.1
通讯作者：
Hung-Yuan Fan;Peter Chang-Yi Weng;E. Chu
中科院分区：
文献类型：
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作者：
Hung-Yuan Fan;Peter Chang-Yi Weng;E. Chu
我们考虑的数值解的广义李雅普诺夫和斯坦方程，分别产生于连续和离散时间的随机最优控制。推广的史密斯方法，我们的算法二次收敛，每次迭代的计算复杂度为O（n3），内存需求为O（n2）。对于大规模问题，当相关的矩阵算子是“稀疏的”时，我们对广义Stein（或Lyapunov）方程的算法可以达到O（n）的复杂度和内存要求（或类似于与稀疏矩阵算子相关的线性系统的解）.这些有效的算法可以应用于牛顿法求解有理Riccati方程。这与复杂度为O（n6）的朴素牛顿算法或复杂度为O（n3）的较慢的修改牛顿方法形成了有利的对比。收敛性和误差分析将被考虑和数值例子提供。
We consider the numerical solution of the generalized Lyapunov and Stein equations in, arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have anO(n3) computational complexity per iteration and anO(n2) memory requirement. For large-scale problems, when the relevant matrix operators are “sparse”, our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement ofO(n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton’s method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms ofO(n6) complexity or the slower modified Newton’s methods ofO(n3) complexity. The convergence and error analysis will be considered and numerical examples provided.