有理 Krylov 子空间算法的最优参数选取

The rational Krylov subspace method is one of the most important methods in dealing with large-scale problems. It has a wide range of applications, such as eigenvalue problems, model reduction problems and matrix equations. Whether the method is successful largely depends on the choice of the parameters. Therefore, it is necessary and important to explore the optimization theory for the parameters. In this project, we do the following researches: firstly, we set up a theory on the optimal choice of the parameters for the rational Krylov subspace methods，which are used for solving the algebraic Riccati equations and some model reduction problems. Secondly, we put forward the relations between the parameters and the Ritz values from the projection matrix. It is also analyzed how these values influence the convergence rate of the rational Krylov subspace method. Finally, we devise new algorithms for obtaining the optimal parameters. Numerical experiments are done to verify the validity of the theoretical analysis and to illustrate the advantage of the new algorithms.

有理 Krylov 子空间算法是求解大规模矩阵问题的一种重要算法。它在特征值问题,模型降阶问题,矩阵方程求解等方面都有广泛的应用。有理 Krylov 子空间算法能否成功在很大程度上取决于参数的选取是否合适。因此，研究最优参数的选取理论和快速算法是十分必要的。本项目拟做如下的研究工作：针对代数 Riccati 方程以及几类模型降阶问题的有理 Krylov 子空间算法,建立最优参数的选取理论；研究参数与投影矩阵的 Ritz 值之间的关系问题，分析参数和 Ritz 值的变化将如何影响有理 Krylov 子空间算法的收敛速度；设计新的最优参数选取算法。在本项目中，会进行大量数值实验来表明理论结果的正确性以及新的选取算法的高效性。