Abstract
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.
Highlights
The basis for numerous practical applications and research areas, such as systems and control theory, consists of modeling large-scale technical and dynamical systems
These results allow us to formulate the final version of the low-rank alternating direction implicit (LR-ADI) iteration for standard Lyapunov equations, which we summarize in Algorithm 3
The LR-ADI iteration is an efficient approach to approximate the solution of Lyapunov equations, which benefits from low-rank formulations for the solution and residual
Summary
The basis for numerous practical applications and research areas, such as systems and control theory, consists of modeling large-scale technical and dynamical systems. Due to the growing complexity of modern applications, the occurrence of systems of differential equations that are too large for numerical computations or simulations is not uncommon To overcome this issue, a variety of model order reduction methods have been developed. When dealing with small Lyapunov equations, it is appropriate to use direct solvers like the Bartels-Stewart method [6] and Hammarling’s algorithm [15], which are based on the Schur decomposition of the system matrix.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
