Low rank methods for a class of generalized Lyapunov equations and related issues

Abstract

In this paper, we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation, we can even extend the result to the $$d$$ -dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krylov-Plus-Inverted-Krylov (K-PIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction, we will show the efficiency of the new methods.