大型结构性代数 Riccati 方程的高效求解算法研究

This project focuses on developing efficient algorithms for solving large-scale structured algebraic Riccati equations (AREs), which are not only important on numerical algebra with great theoretical significance, but also have valuable applications in physical theory calculation, control theory and many other fields. Although they are important and have extensive applications, there are rare efficient algorithms to solve such problems, due to the essential difficulties of the problem. In this project, we consider the large-scale structured AREs in three aspects: 1) by deeply analyzing the iteration formula of the structure-preserving doubling algorithm for medium-size AREs, we propose to develop decoupled algorithms to solve large-scale AREs with low-rank structure, and then make robust analysis and error analysis for the novel algorithms; 2) based on the high accuracy requirement on the solutions to M-matrix AREs, we will combine the high-precision calculation technique with the proposed decoupled algorithm and finally produce a package applied to practical applications; 3) to consummate the convergence analysis of the structure-preserving doubling algorithm, we also propose to prove its convergence at the critical case by functional analysis and operator theory.

本项目探讨大型结构性代数Riccati方程的高效求解算法．大型结构性代数Riccati方程的高效求解问题不仅是个重要的数值代数问题，有着重要的理论意义，而且在物理理论计算、现代控制理论等领域中有着举足轻重的应用价值．虽然大型结构性代数Riccati方程有着广泛的应用背景，但由于问题本身的困难性，目前仍缺少可行有效算法．本项目拟从以下三个方面考虑大型结构性代数Riccati方程的求解问题：通过对求解中小型代数Riccati方程的保结构算法的迭代公式的深入分析，发展求解大型具有低秩结构的代数Riccati方程的解耦型算法，并对算法进行稳定性分析和误差分析；针对M-矩阵代数Riccati方程对解的高精度要求，探求将高精度计算的技巧与解耦型算法相结合的求解方法，力图发展能够求解实际应用问题的软件包；为完善保结构加倍算法的收敛性分析，从泛函分析和算子理论角度探索保结构算法在临界情形下的收敛性证明．

代数Riccati方程的高效求解问题不仅是个重要的数值代数问题，有着重要的理论意义，而且在物理理论计算、现代控制理论等领域中有着举足轻重的应用价值。本项目探讨大型结构性代数Riccati方程的高效求解算法，揭示了问题内蕴的结构，发展了相应的保结构类型算法，建立了具有普适性的分析框架。具体地，本项目取得了如下成果：（1）针对问题的低秩结构，本项目发展了解耦类型的保结构算法，保持了原问题的低秩结构，避免矩阵求逆或者右端项为矩阵的线性系统的求解；（2）利用矩阵分析和不动点迭代的手法给出了状态矩阵A处于不稳定状态下的方程唯一稳定解的闭形式，挖掘了其内蕴的Toeplitz结构，发展了基于快速Fourier变换的全新稳定算法；（3）通过半张量积的引入，将确定性问题的分析框架拓展到了随机代数Riccati方程上，研究了方程及其唯一稳定解的代数结构并设计了相应的求解算法。