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Author

Jianyu Pan

Other affiliations: Chinese Academy of Sciences
Bio: Jianyu Pan is an academic researcher from East China Normal University. The author has contributed to research in topics: Toeplitz matrix & Preconditioner. The author has an hindex of 13, co-authored 18 publications receiving 1016 citations. Previous affiliations of Jianyu Pan include Chinese Academy of Sciences.

Papers
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Journal ArticleDOI
TL;DR: A class of preconditioned Hermitian/skew-Hermitian splitting iteration methods is established, showing that the new method converges unconditionally to the unique solution of the linear system.
Abstract: For the positive semidefinite system of linear equations of a block two-by-two structure, by making use of the Hermitian/skew-Hermitian splitting iteration technique we establish a class of preconditioned Hermitian/skew-Hermitian splitting iteration methods. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.

465 citations

Journal ArticleDOI
TL;DR: New preconditioners based on matrix splittings for the saddle point problems are presented and the spectral property of one of the preconditionsed matrix is studied in detail.

139 citations

Journal ArticleDOI
TL;DR: An approximate inverse preconditioner is constructed to approximate the inverses of weighted Toeplitz matrices by circulant matrices, and then combine them together row-by-row, and it is shown that the spectra of the resulting preconditionsed matrices are clustered around one.
Abstract: The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grunwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners may not work for such Toeplitz-like linear systems. The main aim of this paper is to propose and develop approximate inverse preconditioners for such Toeplitz-like matrices. An approximate inverse preconditioner is constructed to approximate the inverses of weighted Toeplitz matrices by circulant matrices, and then combine them together row-by-row. Because of Toeplitz structure, both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast Fourier transforms. Theoretically, we show that the spectra of the resulting preconditioned matrices are clustered around one. Thus Krylov subspace methods with the proposed preconditione...

117 citations

Journal ArticleDOI
TL;DR: A more general choice of the LM parameter is given, and it is shown that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity.

60 citations

Journal ArticleDOI
TL;DR: Under the local error bound condition which is weaker than the nonsingularity, it is shown that the Levenberg-Marquardt method converges superlinearly to the solution for δ∈ (0, 1), while quadratically for η∈ [1, 2].
Abstract: We propose a new self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear equations F(x) = 0. The Levenberg-Marquardt parameter is chosen as the product of ?Fk?? with ? being a positive constant, and some function of the ratio between the actual reduction and predicted reduction of the merit function. Under the local error bound condition which is weaker than the nonsingularity, we show that the Levenberg-Marquardt method converges superlinearly to the solution for ?? (0, 1), while quadratically for ?? [1, 2]. Numerical results show that the new algorithm performs very well for the nonlinear equations with high rank deficiency.

54 citations


Cited by
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Book ChapterDOI
01 Jan 2015

3,828 citations

01 Jan 2007
TL;DR: An upper bound of the contraction factor of the HSS iteration is derived which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved.
Abstract: We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.

760 citations

Journal ArticleDOI
TL;DR: A preconditioning strategy based on the symmetric\slash skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditionsed matrix are established.
Abstract: In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric\slash skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical experiments with matrices from various application areas.

412 citations

Journal ArticleDOI
TL;DR: A generalized SOR (GSOR) method is presented to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration.
Abstract: For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71---85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method.

347 citations

Journal ArticleDOI
TL;DR: A new splitting is introduced, called positive-definite and skew-Hermitian splitting (PSS), and a class of PSS methods similar to the HSS and NSS method for iteratively solving the positive- definite systems of linear equations are established.
Abstract: By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian splitting (PSS), and then establish a class of PSS methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS) method for iteratively solving the positive-definite systems of linear equations. Theoretical analysis shows that the PSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrix and independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. When we specialize the PSS to block triangular (or triangular) and skew-Hermitian splitting (BTSS or TSS), the PSS method naturally leads to a BTSS or TSS iteration method, which may be more practical and efficient than the HSS and NSS iteration methods. Applications of the BTSS method to the linear systems of block two-by-two structures are discussed in detail. Numerical experiments further show the effectiveness of our new methods.

304 citations