scholarly journals A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations

2012 ◽  
Vol 436 (8) ◽  
pp. 2807-2827 ◽  
Author(s):  
Miloud Sadkane
Keyword(s):  
Discrete Time ◽  
Large Scale ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Smith Method

scholarly journals Efficient Techniques for Solving the Periodic Projected Lyapunov Equations and Model Reduction of Periodic Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mohammad-Sahadet Hossain ◽  
M. Monir Uddin
Keyword(s):  
Model Reduction ◽  
Discrete Time ◽  
Large Scale ◽  
Computational Cost ◽  
Descriptor Systems ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Adi Method ◽  
Smith Method ◽  
Alternating Direction

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.

The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

2021 ◽  
Vol 8 (3) ◽  
pp. 526-536
Author(s):  
L. Sadek ◽  
◽  
H. Talibi Alaoui ◽  
Keyword(s):  
Matrix Equation ◽  
Large Scale ◽  
Krylov Subspace ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Lanczos Algorithm ◽  
Lyapunov Matrix ◽  
Block Lanczos ◽  
Low Dimensional ◽  
Differential Lyapunov Equations

In this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.

scholarly journals Inexact methods for the low rank solution to large scale Lyapunov equations

2020 ◽  
Vol 60 (4) ◽  
pp. 1221-1259 ◽  
Author(s):  
Patrick Kürschner ◽  
Melina A. Freitag
Keyword(s):  
Large Scale ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Inexact Methods

Large-scale Stein and Lyapunov equations, Smith method, and applications

2012 ◽  
Vol 63 (4) ◽  
pp. 727-752 ◽  
Author(s):  
Tiexiang Li ◽  
Peter Chang-Yi Weng ◽  
Eric King-wah Chu ◽  
Wen-Wei Lin
Keyword(s):  
Large Scale ◽  
Lyapunov Equations ◽  
Smith Method

scholarly journals Low-Rank Alternating Direction Implicit Iteration in pyMOR

2020 ◽  
Vol 2 (1) ◽  
pp. 1-13
Author(s):  
Linus Balicki
Keyword(s):  
Large Scale ◽  
Low Rank ◽  
Balanced Truncation ◽  
Lyapunov Equations ◽  
Software Library ◽  
Alternating Direction Implicit ◽  
Truncation Method ◽  
Alternating Direction ◽  
Adi Iteration ◽  
Implicit Iteration

The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework.

Low-Rank Solution of Lyapunov Equations

SIAM Review ◽  
2004 ◽  
Vol 46 (4) ◽  
pp. 693-713 ◽  
Author(s):  
Jing-Rebecca Li ◽  
Jacob White
Keyword(s):  
Low Rank ◽  
Lyapunov Equations
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