A preconditioned block Arnoldi method for large Sylvester matrix equations

Abstract

In this paper, we propose a block Arnoldi method for solving the continuous low-rank Sylvester matrix equation AX C XB D EF T . We consider the case where both A and B are large and sparse real matrices, and E and F are real matrices with small rank. We first apply an alternating directional implicit preconditioner to our equation, turning it into a Stein matrix equation. We then apply a block Krylov method to the Stein equation to extract low-rank approximate solutions. We give some theoretical results and report numerical experiments to show the efficiency of this method.where the unknown matrix X 2 R n s , the coefficient matrices A 2 R n n , B 2 R s s and E 2 R n r , F 2 R s r are full rank with r n, s. Sylvester equations arise in numerous applied areas such as control and communication theory and model reduction problems [1][2][3]. The matrix Equation (1) appears also in the numerical solution of matrix differential Riccati equations, in decoupling techniques for ordinary and partial differential equations, and in filtering and image restoration (see, e.g., [4][5][6] and also the references [7,8]).The matrix Equation (1) can be reformulated as the ns ns linear system .

A note on the Davison–Man method for Sylvester matrix equations

“…Many variants, based on different choices of the Krylov space: block Krylov (Jaimoukha and Kasenally 1994;El Guennouni et al 2002;Jbilou 2006), ADI-preconditioned block Krylov (Bouhamidi et al 2013), extended block Krylov (Druskin and Knizhnerman 1998;Druskin et al 2011;Heyouni 2010;Simoncini 2007), extended global Arnoldi (Heyouni 2010) have been described in the literature. In the sequel, we give some background on the extended block Arnoldi method (EBA) for of which Heyouni showed the advantages in his paper Heyouni (2010), bearing in mind that another block Krylov subspaces method could be preferred in certain particular cases.…”

“…The Smith method (Penzl 2000;Smith 1968) and methods based on the alternating directional implicit (in short ADI) algorithm (Benner et al 2009;Benner and Kürschner 2014;Hu and Reichel 1992) can also be applied if some information about the spectra of A and B is given. Note that, ADI iterations allow faster convergence over Krylov methods if sub-optimal shifts to A and B can be effectively computed and linear systems with shifted coefficient matrices are solved effectively at low cost (Benner et al 2009;Bouhamidi et al 2013).…”

Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

“…The set of shift parameters is then chosen as a subset of these Ritz values. This procedure is widely used in the ADI-type methods for solving large scale matrix equations such as Lyapunov or Sylvester matrix equations; see for example [3,16] …”

A preconditioned block Arnoldi method for large scale Lyapunov and algebraic Riccati equations

“…At each step of the Newton method, we solve a large Lyapunov matrix equation. To get approximate solutions to this matrix equation, we use the block Arnoldi algorithm associated with a preconditioner based on the alternating direction implicit (ADI) iteration method [4,20,22,29]. This will allow us to obtain a rapid convergence and this reduces the costs in terms of storage and cpu-time.…”