Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices

Bai, Zhong‐Zhi; Golub, Gene H.; Li, Chi-Kwong

doi:10.1137/050623644

Cited by 155 publications

(72 citation statements)

References 19 publications

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“…In addition, the upper bound of its asymptotic convergence rate is only dependent on the spectrum of the Hermitian part H, but is independent of the spectrum of the skew-Hermitian part S. To learn more about the HSS method and its variants, one can refer to references [7,10,11,8] for system of real linear equations, references [2,3] for system of complex linear equations, and references [1,4,5,6,12] for system of linear equations with block two-by-two coefficient matrix.…”

Section: The Hss Methodsmentioning

confidence: 99%

The WR–HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

Yang

2017

Journal of Computational and Applied Mathematics

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We consider the numerical method for non-self-adjoint positive definite linear differential equations, and its application to the unsteady discrete elliptic problem, which is derived from spatial discretization of the unsteady elliptic problem with Dirichlet boundary condition. Based on the idea of the alternating direction implicit (ADI) iteration technique and the Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation (WR) iteration method for solving the non-self-adjoint positive definite linear differential equations, called the WR-HSS method. We analyze the convergence property of the WR-HSS method, and prove that the WR-HSS method is unconditionally convergent to the solution of the system of linear differential equations. In addition, we derive the upper bound of the contraction factor of the WR-HSS method in each iteration which is only dependent on the Hermitian part of the corresponding non-self-adjoint positive definite linear differential operator. Finally, the applications of the WR-HSS method to the unsteady discrete elliptic problem demonstrate its effectiveness and the correctness of the theoretical results.

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Section: The Hss Methodsmentioning

confidence: 99%

The WR–HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

Yang

2017

Journal of Computational and Applied Mathematics

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“…Assume that (x * , y * ) * is the eigenvector corresponding to the eigenvalue λ of the iteration matrix T 1 (ω, τ, γ ) defined as in (6) with M(ω) defined by (5).…”

Section: Lemmamentioning

confidence: 99%

“…The matrix M (ω, τ, γ ) is defined as in (2) with M(ω) defined by (5). Then the APIU-SSOR method for nonsingular saddle-point problem (1) is convergent if the iteration parameters ω, τ and γ satisfy:…”

Section: Lemmamentioning

confidence: 99%

“…M(ω) is defined as in (4) or (5). Then V (T 2 (ω, τ, γ )) < 1 if ρ(T (ω, τ, γ )) < 1 with T (ω, τ, γ ) being defined by (16).…”

Section: Lemma 5 ([19])mentioning

confidence: 99%

“…For example, the matrix splitting iterative methods [7,11,18,36,40,45], Uzawa-type methods [12,17,21,22,25], HSS method and its variants [2,[4][5][6]8,9,28,29,38], Krylov subspace methods [1,10,[33][34][35]42], null space methods [24] and so on. When the saddle-point problem (1) is singular, there are also many relaxation iteration methods which have been established, e.g., the Uzawa-type methods [30,46,48,49] and the HSS-like methods [3,20,39].…”

Section: Introductionmentioning

confidence: 99%

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Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

Liang

Zhang

2015

Bit Numer Math

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In this paper, based on the SOR and SSOR splittings of the (1,1) part of saddle-point coefficient matrix, some variants of the accelerated parameterized inexact Uzawa (VAPIU) method are proposed for solving nonsingular and singular saddlepoint problems. By choosing different parameter matrices, we derive some existing and new iterative methods. The corresponding convergence and semi-convergence of the VAPIU methods for solving nonsingular and singular saddle-point problems are studied in depth, respectively. The preconditioning strategies based on the VAPIU splittings of the coefficient matrices are presented. Numerical experiments are provided, which confirms that these new methods need less CPU times per iteration step comparing with some other methods for solving both nonsingular and singular saddle-point problems.

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The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

2017

Asian Journal of Control

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In this paper, the unified frame of alternating direction method of multipliers (ADMM) is proposed for solving three classes of matrix equations arising in control theory including the linear matrix equation, the generalized Sylvester matrix equation and the quadratic matrix equation. The convergence properties of ADMM and numerical results are presented. The numerical results show that ADMM tends to deliver higher quality solutions with less computing time on the tested problems.

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Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices

Cited by 155 publications

References 19 publications

The WR–HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

The WR–HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

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