Research Initiation Award: Numerical Solution of Large-Scale Algebraic Riccati Equations

提出了求解超大矩阵代数Riccati方程的并行和矢量算法的发展。要使用的方法强调使用诸如BLAS3之类的计算内核，这些内核可用于各种并行计算机。这种方法将使算法可移植到许多先进的科学计算机上。许多研究都是基于矩阵符号函数算法，这是一种迭代方法，已被证明在某类多处理器上工作良好。研究主题包括易于并行化的“富乘法”算法，几种不同先进计算机体系结构的并行和矢量算法，使用矩阵符号函数估计可计算的条件，以及求解线性李雅普诺夫方程的迭代算法。最后，提出了将这些算法作为高质量、可靠的数学软件来实现，以便使研究结果可供控制系统界使用。代数Riccati方程的求解是控制工程中的一个基本计算问题，对许多现代控制技术的应用至关重要。

The development of parallel and vector algorithms for solving very large matrix algebraic Riccati equations is proposed. The approach to be used stresses the use of computational kernels such as the BLAS3, which are available on a wide variety of parallel computers. This approach will make the algorithms portable to many advanced scientific computers. Much of the research is based on the matrix sign function algorithm, an iterative method that has been shown to work well on a certain class of multiprocessor. Topics for research include easily parallelized "multiplication rich" algorithms, parallel and vector algorithms for several different advanced computer architectures, a condition estimate computable using the matrix sign function, and iterative algorithms for solving the linear Lyapunov equation. Finally, implementation of the algorithms as high quality, reliable mathematical software is proposed, in order to make the results of the research available for use by the control systems community. The solution of algebraic Riccati equations is a fundamental computational problem in control engineering and is essential for the application of many modern control techniques.