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。AI 中文总结
本文提出一种低秩ADI算法，用于高效求解大规模离散时间非对称代数Riccati方程，通过递归构造低秩镇定解并自动生成移位，在10^6至10^7维问题上验证了准确性和效率。
AI 中文摘要
离散时间非对称代数Riccati方程（DTNAREs）出现在纳什均衡的博弈论计算中。大规模求解此类方程通常在计算上是不可行的。本文针对解具有低秩结构的大规模DTNAREs，提出了一种数值方法。文中引入了一种低秩交替方向隐式（ADI）方法，该方法递归地构造低秩镇定解，而无需显式求解任何投影DTNARE。通过低秩ADI迭代的极点配置性质，该方法确保了隐式求解的投影DTNARE始终存在镇定解。此外，还开发了一种用于ADI迭代的自动移位生成策略。一旦提供了初始移位，算法无需用户进一步干预即可计算低秩解。在维度介于10^6和10^7之间的DTNAREs上进行的数值实验证明了该方法的准确性和效率。结果证实，所提出的低秩ADI算法是求解大规模DTNAREs的有效求解器，而这些方程在其他情况下在计算上是不可行的。
英文摘要
Discrete-time non-symmetric algebraic Riccati equations (DTNAREs) arise in game-theoretic computations of Nash equilibria. Solving such equations at large scale is often computationally prohibitive. This paper develops a numerical approach for large-scale DTNAREs whose solutions are low rank. A low-rank alternating direction implicit (ADI) method is introduced that recursively constructs a low-rank stabilizing solution without explicitly solving any projected DTNARE. Through the pole-placement property of the low-rank ADI iteration, the method ensures that the implicitly solved projected DTNARE always admits a stabilizing solution. An automatic shift-generation strategy is also developed for the ADI iterations. Once an initial shift is provided, the algorithm computes the low-rank solution without further user intervention. Numerical experiments on DTNAREs with dimensions between \(10^6\) and \(10^7\) demonstrate the accuracy and efficiency of the method. The results confirm that the proposed low-rank ADI algorithm is an effective solver for large-scale DTNAREs that would otherwise be computationally prohibitive.
