Solution of Large - Scale Lyapunov Equations via the Block Modified Smith Method

Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method by John Sabino Balanced truncation is an attractive method for reducing the dimension of mediumscale dynamical systems. Research in recent years has brought approximate balanced truncation to the large-scale setting. At the heart of this technique are alternating direction implicit (ADI) methods for solving large Lyapunov and Sylvester equations. This work concerns the convergence of these methods. Our primary objective is the practical solution of very large Lyapunov equations. Uncertainty in the selection of shifts for the ADI method and its variants has prevented the widespread adoption of an otherwise promising variant, the block modified Smith method. We examine in detail the role of shift selection, the fundamental minimax problem, and the often startling influence of nonnormality. Our analysis is tied intimately to the decay rate of the singular values of the solutions to these equations. We improve upon past bounds and develop new ones, including bounds based on pseudospectra. In the end, we provide simple yet effective schemes for finding shifts that outperform those produced by conventional selection strategies. To make effective use of these shifts, we provide insights that allow the block modified Smith method to be applied to very large equations in a practical setting. First, we give an error bound in terms of the drop tolerance that substantially improves upon ad hoc choices that could fail or demand excessive memory. Next, we improve the residual calculation in the method, which can be a surprisingly expensive computation. Finally, we provide guidance on the use of complex shifts, when to update the singular value decomposition, and the number of shifts to use. We have succeeded in providing a practical implementation of the block modified Smith method. The success of the algorithm is demonstrated in numerous experiments, culminating in the rapid solution of some of the largest known Lyapunov equations attempted on a workstation.