• Title, Summary, Keyword: saddle point

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A GENERALIZATION OF LOCAL SYMMETRIC AND SKEW-SYMMETRIC SPLITTING ITERATION METHODS FOR GENERALIZED SADDLE POINT PROBLEMS

  • Li, Jian-Lei;Luo, Dang;Zhang, Zhi-Jiang
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1167-1178
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    • 2011
  • In this paper, we further investigate the local Hermitian and skew-Hermitian splitting (LHSS) iteration method and the modified LHSS (MLHSS) iteration method for solving generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. When A is non-symmetric positive definite, the convergence conditions are obtained, which generalize some results of Jiang and Cao [M.-Q. Jiang and Y. Cao, On local Hermitian and Skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 2009(231): 973-982] for the generalized saddle point problems to generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. Numerical experiments show the effectiveness of the iterative methods.

SEMI-CONVERGENCE OF THE PARAMETERIZED INEXACT UZAWA METHOD FOR SINGULAR SADDLE POINT PROBLEMS

  • YUN, JAE HEON
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1669-1681
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    • 2015
  • In this paper, we provide semi-convergence results of the parameterized inexact Uzawa method with singular preconditioners for solving singular saddle point problems. We also provide numerical experiments to examine the effectiveness of the parameterized inexact Uzawa method with singular preconditioners.

Location of Transition States by the Conjugate Reaction Coordinate Method

  • Lee, Ik-Choon;Lee, Bon-Su;Kim, Chan-Kyung
    • Bulletin of the Korean Chemical Society
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    • v.7 no.5
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    • pp.376-379
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    • 1986
  • A relatively simple method of locating the saddle point is presented. In this method a single determination of the saddle point location by constrained energy minimizations for points selected on the assumed saddle surface provides us with the structure, location and energy of the TS, the reaction path at the saddle point and characterization as the TS. Some examples were given.

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A PARALLEL IMPLEMENTATION OF A RELAXED HSS PRECONDITIONER FOR SADDLE POINT PROBLEMS FROM THE NAVIER-STOKES EQUATIONS

  • JANG, HO-JONG;YOUN, KIHANG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.22 no.3
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    • pp.155-162
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    • 2018
  • We describe a parallel implementation of a relaxed Hermitian and skew-Hermitian splitting preconditioner for the numerical solution of saddle point problems arising from the steady incompressible Navier-Stokes equations. The equations are linearized by the Picard iteration and discretized with the finite element and finite difference schemes on two-dimensional and three-dimensional domains. We report strong scalability results for up to 32 cores.

ON THE GENERALIZED SOR-LIKE METHODS FOR SADDLE POINT PROBLEMS

  • Feng, Xin-Long;Shao, Long
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.663-677
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    • 2010
  • In this paper, the generalized SOR-like methods are presented for solving the saddle point problems. Based on the SOR-like methods, we introduce the uncertain parameters and the preconditioned matrixes in the splitting form of the coefficient matrix. The necessary and sufficient conditions for guaranteeing its convergence are derived by giving the restrictions imposed on the parameters. Finally, numerical experiments show that this methods are more effective by choosing the proper values of parameters.

ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS

  • Yun, Jae Heon
    • Journal of the Korean Mathematical Society
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    • v.53 no.3
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    • pp.691-707
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    • 2016
  • In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.

FAST ONE-PARAMETER RELAXATION METHOD WITH A SCALED PRECONDITIONER FOR SADDLE POINT PROBLEMS

  • OH, SEYOUNG;YUN, JAE HEON
    • Journal of applied mathematics & informatics
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    • v.34 no.1_2
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    • pp.85-94
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    • 2016
  • In this paper, we first propose a fast one-parameter relaxation (FOPR) method with a scaled preconditioner for solving the saddle point problems, and then we present a formula for finding its optimal parameter. To evaluate the effectiveness of the proposed FOPR method with a scaled preconditioner, numerical experiments are provided by comparing its performance with the existing one or two parameter relaxation methods with optimal parameters such as the SOR-like, the GSOR and the GSSOR methods.

ON A SPLITTING PRECONDITIONER FOR SADDLE POINT PROBLEMS

  • SALKUYEH, DAVOD KHOJASTEH;ABDOLMALEKI, MARYAM;KARIMI, SAEED
    • Journal of applied mathematics & informatics
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    • v.36 no.5_6
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    • pp.459-474
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    • 2018
  • Cao et al. in (Numer. Linear. Algebra Appl. 18 (2011) 875-895) proposed a splitting method for saddle point problems which unconditionally converges to the solution of the system. It was shown that a Krylov subspace method like GMRES in conjunction with the induced preconditioner is very effective for the saddle point problems. In this paper we first modify the iterative method, discuss its convergence properties and apply the induced preconditioner to the problem. Numerical experiments of the corresponding preconditioner are compared to the primitive one to show the superiority of our method.

SPECTRAL ANALYSIS OF THE MGSS PRECONDITIONER FOR SINGULAR SADDLE POINT PROBLEMS

  • RAHIMIAN, MARYAM;SALKUYEH, DAVOD KHOJASTEH
    • Journal of applied mathematics & informatics
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    • v.38 no.1_2
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    • pp.175-187
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    • 2020
  • Recently Salkuyeh and Rahimian in (Comput. Math. Appl. 74 (2017) 2940-2949) proposed a modification of the generalized shift-splitting (MGSS) method for solving singular saddle point problems. In this paper, we present the spectral analysis of the MGSS preconditioner when it is applied to precondition the singular saddle point problems with the (1, 1) block being symmetric. Some eigenvalue bounds for the spectrum of the preconditioned matrix are given. We show that all the real eigenvalues of the preconditioned matrix are in a positive interval and all nonzero eigenvalues having nonzero imaginary part are contained in an intersection of two circles.