8 research outputs found
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
Least Squares Based Iterative Algorithm for the Coupled Sylvester Matrix Equations
By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equations AX+YB=C and DX+YE=F in this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equation AXB+CXD=F. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm
Preconditioners based on the Alternating-Direction-Implicit algorithm for the 2D steady-state diffusion equation with orthotropic heterogeneous coefficients
In this paper, we combine the Alternating Direction Implicit (ADI) algorithm with the concept of preconditioning and apply it to linear systems discretized from the 2D steady-state diffusion equations with orthotropic heterogeneous coefficients by the finite element method assuming tensor product basis functions. Specifically, we adopt the compound iteration idea and use ADI iterations as the preconditioner for the outside Krylov subspace method that is used to solve the preconditioned linear system. An efficient algorithm to perform each ADI iteration is crucial to the efficiency of the overall iterative scheme. We exploit the Kronecker product structure in the matrices, inherited from the tensor product basis functions, to achieve high efficiency in each ADI iteration. Meanwhile, in order to reduce the number of Krylov subspace iterations, we incorporate partially the coefficient information into the preconditioner by exploiting the local support property of the finite element basis functions. Numerical results demonstrated the efficiency and quality of the proposed preconditioner. © 2014 Elsevier B.V. All rights reserved
Numerical methods for large-scale Lyapunov equations with symmetric banded data
The numerical solution of large-scale Lyapunov matrix equations with
symmetric banded data has so far received little attention in the rich
literature on Lyapunov equations. We aim to contribute to this open problem by
introducing two efficient solution methods, which respectively address the
cases of well conditioned and ill conditioned coefficient matrices. The
proposed approaches conveniently exploit the possibly hidden structure of the
solution matrix so as to deliver memory and computation saving approximate
solutions. Numerical experiments are reported to illustrate the potential of
the described methods
Matrix Equation Techniques for Certain Evolutionary Partial Differential Equations
We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples
Métodos numéricos para resolução de equações de Lyapunov
O objectivo desta dissertação é descrever, analisar e aplicar alguns métodos numéricos para resolver a equação clássica de Lyapunov.
Estudamos condições que garantem a solubilidade das equações e estabelecemos relações entre a fórmula contínua
AX + X A* + Q = 0
e a fórmula discreta
AX A* − X + Q = 0 .
O produto de Kronecker é usado de modo a permitir representações de equações matriciais e o desenvolvimento de alguns métodos numéricos
Analisamos algumas decomposições matriciais que vão ser utilizadas no desenvolvimento de
alguns métodos numéricos directos nomeadamente Bartels-Stewart e Hessenberg-Schur.
Por fim, os subespaço de Krylov e alguns processos de ortogonalização permitem desenvolver os métodos iterativos de Arnoldi e GMRES e os métodos directos de Ward e Kirrinnis