Parameter modified versions of preconditioning and iterative inner product free refinement methods for two-by-two block matrices

Axelsson, Owe; Liang, Zhao‐Zheng

doi:10.1016/j.laa.2019.07.024

Cited by 4 publications

(2 citation statements)

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“…Therefore, we can use the results in [33,34] to obtain the quasi-optimal parameter α in the following theorem.…”

Section: Algorithm 3 (The Detailed Implementing Process Of the C-to-rmentioning

confidence: 99%

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

Zeng

2020

Mathematics

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To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods.Keywords: weakly nonlinear equations; C-to-R preconditioner; local convergence; Picard iteration; complex symmetric matrix. MSC: 65F10; 65F50; 65N22where A = W + iT ∈ C n×n is a large, sparse, complex symmetric matrix, with W ∈ R n×n and T ∈ R n×n being the real parts and the imaginary parts of the coefficient matrix A, respectively. Here, we assume that W and T are both symmetric positive and semidefinite (SPSD) and at least one of them being symmetric positive and definite (SPD). The right hand vector function φ : D ⊂ C n → C n is a continuously differential function defined on the open convex domain D in the n-dimensional C n . u ∈ C n is an unknown vector. When the linear term Au is strongly dominant over the nonlinear term φ(u) in certain norm [1], we say that the system of nonlinear Equation (1) is weakly nonlinear. Here, and in the sequence, we assume that the Jacobian matrix of the nonlinear function φ(u) at the solution point u ∈ D, denoted as φ (u ), is the non-Hermitian and negative semidefinite.Weakly nonlinear equations of the form (1) arise in many areas of scientific computing and engineering applications, e.g., nonlinear ordinary and partial differential equations, nonlinear integral and integro differential equations, nonlinear optimization and variational problems, saddle point problems from image processing, and so on. For more details, see [2][3][4][5][6][7][8].

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“…Therefore, we can use the results in [33,34] to obtain the quasi-optimal parameter α in the following theorem.…”

Section: Algorithm 3 (The Detailed Implementing Process Of the C-to-rmentioning

confidence: 99%

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

Zeng

2020

Mathematics

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“…Further, an elementary spectral analysis shows that the eigenvalues of B −1 A are contained in the interval [ 1 2 , 1]. As shown in [4], here it is best to use the Chebyshev semi-iteration as acceleration method. This avoids rounding errors as would arise in the vector orthogonalizations if a Krylov subspace method is used.…”

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confidence: 99%

Robust iteration methods for complex systems with an indefinite matrix term

Axelsson¹,

Pourbagher²,

Salkuyeh³

2021

Preprint

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Complex valued systems with an indefinite matrix term arise in important applications such as for certain time-harmonic partial differential equations such as the Maxwell's equation and for the Helmholtz equation. Complex systems with symmetric positive definite matrices can be solved readily by rewriting the complex matrix system in two-by-two block matrix form with real matrices which can be efficiently solved by iteration using the preconditioned square block (PRESB) preconditioning method and preferably accelerated by the Chebyshev method. The appearances of an indefinite matrix term causes however some difficulties. To handle this we propose different forms of matrix splitting methods, with or without any parameters involved. A matrix spectral analyses is presented followed by extensive numerical comparisons of various forms of the methods.

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Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

Axelsson

Karátson

2020

Numer. Math.

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Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.

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Parameter modified versions of preconditioning and iterative inner product free refinement methods for two-by-two block matrices

Cited by 4 publications

References 31 publications

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

Robust iteration methods for complex systems with an indefinite matrix term

Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

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