Speaker: Daniel Szyld
Affiliation: Temple University
Wednesday, 04/03/2020, 11:00, Aula Tonelli, SNS.
Title: Efficient iterative methods for the solution of Generalized
Lyapunov Equations: Block vs. point Krylov projections, and other
controversial decisions.
There has been a flurry of activity in recent years in the area of
solution of matrix equations. In particular, a good understanding has
been reached on how to approach the solution of large scale Lyapunov
equations. An effective way to solve Lyapunov equations of the form
$A^T X + XA + C^T C = 0$, where A and X are n×n, is to use Galerkin
projection with appropriate extended or rational Krylov subspaces.
These methods work in part because the solution is known to be
symmetric positive definite with rapidly decreasing singular values,
and therefore it can be approximated by a low rank matrix X_k = Z_k
Z_k^T. Thus the computations are performed usually with storage which
is lower rank, i.e., much lower than order of n^2 .
Generalized Lyapunov equations have additional terms. In this talk, we
concentrate on equations of the following form
$$
A^T X + XA + \sum_{j = 1}^m N_j X N_j^T + C^T C = 0,
$$
Such equations arise for example in stochastic control.
In the present work, we propose a return to classical iterative
methods, and consider instead stationary iterations. The classical
theory of splittings applies here, and we present a new theorem on the
convervegence when the linear system at each step is solved inexactly.
Several theoretical and computational issues are discussed so as to
make the iteration efficient.
Numerical experiments indicate that this method is competitive vis-à-
vis the current state-of-the-art methods, both in terms of
computational times and storage needs.
This is joint work with Stephen D. Shank and Valeria Simoncini.
https://www.dm.unipi.it/webnew/it/seminari/efficient-iterative-methods-solu…
