A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

Abstract

a b s t r a c tA nested splitting conjugate gradient (NSCG) iterative method and a preconditioned NSCG (PNSCG) iterative method are presented for solving the generalized Sylvester equation with large sparse coefficient matrices, respectively. Both methods are actually inner/outer iterations, which employ the CG-like method as inner iteration to approximate each outer iteration, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrices. Convergence conditions of both methods are studied in depth and numerical experiments demonstrate the efficiency of the proposed methods. Moreover, experimental results show that the PNSCG method is more accurate, robust and effective than the NSCG method.

A general alternating-direction implicit framework with Gaussian process regression parameter prediction for large sparse linear systems

“…In this section, we apply the GADI framework to solve the matrix equation. We use a representative example, i.e., the continuous Sylvester equation, which has been widely used in control theory and numerical PDEs [16,26,28], to demonstrate the implementation. Concretely, the continuous Sylvester equation can be written as…”

“…There are two main methods for selecting parameters. One is traversing parameters or experimental determination within some intervals to obtain relatively optimal parameters [2,16,28]. The advantage of this traversal method is that it can obtain relatively accurate optimal parameters, but obviously, it consumes a lot of extra time.…”

“…The ADI methods are sensitive to splitting matrices and parameters which heavily affect their efficiency. For example, the existing results demonstrate that, when ignoring the consumption of obtaining the optimal splitting parameters, the HSS-type methods can solve sparse linear algebra system of million levels at the best with about hundreds seconds [20], and continuous Sylvester matrix equation at most 256 order with tens to thousands seconds [2,16,26]. It also should be pointed out that the cost of obtaining these optimal splitting parameters in these ADI methods may be much more expensive than solving the linear system itself.…”

An alternating direction method for nonnegative solutions of the matrix equation $$AX+YB=C$$ A X + Y B = C

“…Problem 1 Given A ∈ R m×s , B ∈ R l×n and C ∈ R m×n , find nonnegative matrices X ∈ R s×n and Y ∈ R m×l such that AX + Y B = C. The class of the linear matrix equation has been investigated by many authors and a series of important and useful results has been obtained (Baksalary and Kala 1979;Ziȩtak 1984Ziȩtak , 1988Zak 1985;Yong 2006;Huang 2004;Ke and Ma 2014). For example, Baksalary and Kala (1979) obtained a necessary and sufficient condition for solvability and the general solution of the matrix equation AX − Y B = C. In Ziȩtak (1984), Ziȩtak analyzed and gave an algorithm for the l p solutions of (1.1).…”

An inexact relaxed DPSS preconditioner for saddle point problem

“…And Krylov subspace methods are very slow or even fail to converge if not conveniently preconditioned. Therefore, many researchers devote themselves to the preconditioned iterative methods (1.1) (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). And kinds of preconditioners for saddle point matrix are studied, such as symmetric indefinite preconditioners [28,29], inexact constraint preconditioners [29][30][31][32][33][34] and primal-based penalty preconditioners [35].…”