scholarly journals Fast Positive-Real Balanced Truncation Via Quadratic Alternating Direction Implicit Iteration

2007 ◽  
Vol 26 (9) ◽  
pp. 1725-1731 ◽  
Author(s):  
Ngai Wong ◽  
Venkataramanan Balakrishnan
Keyword(s):  
Balanced Truncation ◽  
Positive Real ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Implicit Iteration

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.

scholarly journals An Alternating Direction Implicit Method for Solving Projected Generalized Continuous-Time Sylvester Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yujian Zhou ◽  
Liang Bao ◽  
Yiqin Lin
Keyword(s):  
Continuous Time ◽  
Computational Cost ◽  
Implicit Method ◽  
Descriptor Systems ◽  
Low Rank ◽  
Balanced Truncation ◽  
Alternating Direction Implicit Method ◽  
Alternating Direction Implicit ◽  
Smith Method ◽  
Alternating Direction

We present the generalized low-rank alternating direction implicit method and the low-rank cyclic Smith method to solve projected generalized continuous-time Sylvester equations with low-rank right-hand sides. Such equations arise in control theory including the computation of inner products andℍ2norms and the model reduction based on balanced truncation for descriptor systems. The requirements of these methods are moderate with respect to both computational cost and memory. Numerical experiments presented in this paper show the effectiveness of the proposed methods.

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