控制理论和计算中一些问题的投影方法
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摘要
本文研究控制理论和计算中一些大规模问题的投影方法。我们给出迭代求解广义Sylvester方程的Galerkin方法和极小残量方法，这两种方法都利用Arnoldi过程构造某个Krylov子空间的一组正交规范基并利用系数矩阵的结构减少了对存储量的需求。基于全局Arnoldi过程，我们又分别给出求解大规模Sylvester方程和大规模广义Sylvester方程的新的投影方法。为求解二次特征值问题，我们首先引入基于方阵A_1和A_2以及具有正交规范列的矩阵Q_1的块二阶Krylov子空间，然后再分别给出生成该子空间一组正交规范基的块二阶Arnoldi过程和生成双正交基的块二阶双正交过程。利用投影技巧给出两种块二阶Krylov子空间方法，这两种方法都直接应用于二次特征值问题，从而保留了问题的结构和性质。然后我们给出保结构模型降阶算法来求解大规模二阶多输入多输出动力系统的模型降阶问题。这是基于块二阶Krylov子空间的投影方法，利用块二阶Arnoldi过程来生成投影空间的一组正交规范基。计算所得的约化系统保留了原始系统的二阶结构。最后，我们给出求解传输理论中非对称代数Riccati方程的修改的简单迭代法和修改的牛顿法。
The present Ph.D. dissertation is concerned with projection methods for large-scale problems in control theory and computation. We propose Galerkin method and minimal residual method for iteratively solving generalized Sylvester equations. The algorithms use Krylov subspace for which orthogonal basis are generated by the Arnoldi procedure and reduce the storage space required by using the structure of the matrix. We propose a new projection method based on global Arnoldi procedure for solving large Sylvester equations and the large generalized Sylvester equations. For solving large-scale quadratic eigenvalue problem (QEP), we first introduce a block second-order Krylov subspace based on a pair of square matrices A_1 and A_2 and an orthonormal matrix Q_1, then we present a block second-order Arnoldi procedure for generating an orthonormal basis of the space and a block second-order biorthogonalization procedure for generating biorthonormal basis. By applying the projection techniques, we derive two block second-order Krylov subspace methods. These methods are applied to the QEP directly. Hence they preserve essential structures and properties of the QEP. We present a structure-preserving model-order reduction method for solving large-scale second-order multi-input multi-output dynamical systems. It is a projection method based on a block second-order Krylov subspace. We use the block second-order Arnoldi procedure to generate an orthonormal basis of the projection subspace. The reduced system preserves the second-order structure of the original system. Finally, we propose a modified simple iterative method and a modified Newton method for nonsymmetric algebraic Riccati equations arising in transport theory.
The present Ph.D. dissertation is concerned with projection methods for large-scale problems in control theory and computation. We propose Galerkin method and minimal residual method for iteratively solving generalized Sylvester equations. The algorithms use Krylov subspace for which orthogonal basis are generated by the Arnoldi procedure and reduce the storage space required by using the structure of the matrix. We propose a new projection method based on global Arnoldi procedure for solving large Sylvester equations and the large generalized Sylvester equations. For solving large-scale quadratic eigenvalue problem (QEP), we first introduce a block second-order Krylov subspace based on a pair of square matrices A_1 and A_2 and an orthonormal matrix Q_1, then we present a block second-order Arnoldi procedure for generating an orthonormal basis of the space and a block second-order biorthogonalization procedure for generating biorthonormal basis. By applying the projection techniques, we derive two block second-order Krylov subspace methods. These methods are applied to the QEP directly. Hence they preserve essential structures and properties of the QEP. We present a structure-preserving model-order reduction method for solving large-scale second-order multi-input multi-output dynamical systems. It is a projection method based on a block second-order Krylov subspace. We use the block second-order Arnoldi procedure to generate an orthonormal basis of the projection subspace. The reduced system preserves the second-order structure of the original system. Finally, we propose a modified simple iterative method and a modified Newton method for nonsymmetric algebraic Riccati equations arising in transport theory.
引文
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[16] L. Bao, Y. Lin and Y. Wei, Krylov subspace methods for the generalized Sylvester equation. Appl. Math. Comput., 175 (2006), pp. 557-573.
[17] L. Bao, Y. Lin and Y. Wei, A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory, Appl. Math. Comput., 181 (2006), pp. 1499-1504.
[18] L. Bao, Y. Lin and Y. Wei, A new projection method for solving large Sylvester equations, Appl. Numer. Math., 57 (2007), pp. 521-532.
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[24] D. Calvetti, B. Lewis and L. Reichel, On the solution of large Sylvester-observer equation, Numer. Linear Algebra Appl., 8 (2001), pp. 435-451.
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[37] R. W. Freund, Model reduction methods based on Krylov susbspace, Acta Numerica, 12 (2003), pp. 267-319.
[38] R. W. Freund, Pade-type model reduction of second-order and high-order linear dynamical systems, Technical report, Department of Mathematics, University of California, Davis, 2004.
[39] R. W. Freund, Krylov subspaces associated with higher-order linear dynamical systems, BIT, 45 (2005), pp. 495-516.
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