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15 May 2014, Volume 32 Issue 3
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Original Articles
A NEW PRECONDITIONING STRATEGY FOR SOLVING A CLASS OF TIMEDEPENDENT PDECONSTRAINED OPTIMIZATION PROBLEMS
Minli Zeng, Guofeng Zhang
2014, 32(3): 215232. DOI:
10.4208/jcm.1401CR3
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MULTIGRID METHOD FOR FLUID DYNAMIC PROBLEMS
Galina Muratova, Evgenia Andreeva
2014, 32(3): 233247. DOI:
10.4208/jcm.1403CR11
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152
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This paper covers the review and some aspects of using Multigrid method for fluid dynamics problems. The main development stages of multigrid technics are presented. Some approaches for solving NavierStokes equations and convectiondiffusion problems are considered.
GENERALIZED CONJUGATE AORTHOGONAL RESIDUAL SQUARED METHOD FOR COMPLEX NONHERMITIAN LINEAR SYSTEMS
Jianhua Zhang, Hua Dai
2014, 32(3): 248265. DOI:
10.4208/jcm.1401CR13
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Recently numerous numerical experiments on realistic calculation have shown that the conjugate Aorthogonal residual squared (CORS) method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irregular convergence, especially appears large intermediate residual norm, which may lead to worse approximate solutions and slower convergence rate. In this paper, we present a new producttypemethod for solving complex nonHermitian linear systems based on the biconjugate Aorthogonal residual (BiCOR) method, where one of the polynomials is a BiCOR polynomial, and the other is a BiCOR polynomial with the same degree corresponding to different initial residual. Numerical examples are given to illustrate the effectiveness of the proposed method.
USING THE SKEWSYMMETRIC ITERATIVE METHODS FOR SOLUTION OF AN INDEFINITE NONSYMMETRIC LINEAR SYSTEMS
B.L. Krukier, L.A. Krukier
2014, 32(3): 266271. DOI:
10.4208/jcm.1403CR2
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The concept of the field of value to localize the spectrum of the iteration matrices of the skewsymmetric iterative methods is further exploited. Obtained formulas are derived to relate the fields of values of the original matrix and the iteration matrix. This allows us to determine theoretically that indefinite nonsymmetric linear systems can be solved by this class of iterative methods.
ON BLOCK PRECONDITIONERS FOR PDECONSTRAINED OPTIMIZATION PROBLEMS
Xiaoying Zhang, Yumei Huang
2014, 32(3): 272283. DOI:
10.4208/jcm.1401CR4
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Recently, Bai proposed a blockcounterdiagonal and a blockcountertriangular preconditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finiteelement discretizations of the distributed control problems in (Computing 91 (2011) 379395). He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the blockcounterdiagonal and the blockcountertriangular preconditioners. Experimental results show that the convergence analyses match well with the numerical results.
PARALLEL QUASICHEBYSHEV ACCELERATION TO NONOVERLAPPING MULTISPLITTING ITERATIVE METHODS BASED ON OPTIMIZATION
Ruiping Wen, Guoyan Meng, Chuanlong Wang
2014, 32(3): 284296. DOI:
10.4208/jcm.1401CR1
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In this paper, we present a parallel quasiChebyshev acceleration applied to the nonoverlapping multisplitting iterative method for the linear systems when the coefficient matrix is either an
H
matrix or a symmetric positive definite matrix. First,
m
parallel iterations are implemented in
m
different processors. Second, based on
l
_{1}
norm or
l
_{2}
norm, the
m
optimization models are parallelly treated in
m
different processors. The convergence theories are established for the parallel quasiChebyshev accelerated method. Finally, the numerical examples show that the parallel quasiChebyshev technique can significantly accelerate the nonoverlapping multisplitting iterative method.
ON AUGMENTED LAGRANGIAN METHODS FOR SADDLEPOINT LINEAR SYSTEMS WITH SINGULAR OR SEMIDEFINITE (1, 1) BLOCKS
Tatiana S. Martynova
2014, 32(3): 297305. DOI:
10.4208/jcm.1401CR7
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An effective algorithm for solving large saddlepoint linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skewHermitian triangular splitting iteration methods. We consider the saddlepoint linear systems with singular or semidefinite (1, 1) blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce highquality preconditioners for the Krylov subspace methods for solving large sparse saddlepoint linear systems.
ALTERNATELY LINEARIZED IMPLICIT ITERATION METHODS FOR SOLVING QUADRATIC MATRIX EQUATIONS
Bing Gui, Hao Liu, Minli Yan
2014, 32(3): 306311. DOI:
10.4208/jcm.1401CR9
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A numerical solution of the quadratic matrix equations associated with a nonsingular Mmatrix by using the alternately linearized implicit iteration method is considered. An iteration method for computing a nonsingular Mmatrix solution of the quadratic matrix equations is developed, and its corresponding theory is given. Some numerical examples are provided to show the efficiency of the new method.
THE GENERALIZED LOCAL HERMITIAN AND SKEWHERMITIAN SPLITTING ITERATION METHODS FOR THE NONHERMITIAN GENERALIZED SADDLE POINT PROBLEMS
Hongtao Fan, Bing Zheng
2014, 32(3): 312331. DOI:
10.4208/jcm.1401CR6
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For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skewHermitian splitting iteration methods for nonHermitian saddle point problem [M.Z. Zhu, Appl. Math. Comput. 218 (2012) 8816–8824 ]. In this paper, we further investigate the generalized local Hermitian and skewHermitian splitting (GLHSS) iteration methods for solving nonHermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the convergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.
A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVERPENALIZED INTERIOR PENALTY METHOD FOR NONSELFADJOINT AND INDEFINITE PROBLEMS
Yuping Zeng, Jinru Chen, Feng Wang, Yanxia Meng
2014, 32(3): 332347. DOI:
10.4208/jcm.1403CR5
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207
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In this paper, we study a weakly overpenalized interior penalty method for nonselfadjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residualbased a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.
COMPACT DIFFERENCE SCHEMES FOR THE DIFFUSION AND SCHR¨ ODINGER EQUATIONS. APPROXIMATION, STABILITY, CONVERGENCE, EFFECTIVENESS, MONOTONY
Vladimir A. Gordin, Eugeny A. Tsymbalov
2014, 32(3): 348370. DOI:
10.4208/jcm.1403CR14
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Various compact difference schemes (both old and new, explicit and implicit, onelevel and twolevel), which approximate the diffusion equation and Schr¨odinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.
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