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11
Optimization Of The Hermitian And SkewHermitian Splitting Iteration For SaddlePoint Problems
, 2003
"... We study the asymptotic rate of convergence of the alternating Hermitian/skewHermitian iteration for solving saddlepoint problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal con ..."
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Cited by 22 (13 self)
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We study the asymptotic rate of convergence of the alternating Hermitian/skewHermitian iteration for solving saddlepoint problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in divgrad form. We show that the optimized convergence rate for small mesh parameter h is asymptotically 1  O(h ). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, hindependent, convergence rate. The theoretical analysis is supported by numerical experiments.
Block preconditioning of realvalued iterative algorithms for complex linear systems
, 2008
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CONVERGENCE PROPERTIES OF PRECONDITIONED HERMITIAN AND SKEWHERMITIAN SPLITTING METHODS FOR NONHERMITIAN POSITIVE SEMIDEFINITE MATRICES
"... Abstract. For the nonHermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skewHermitian splitting iteration methods. We then apply these results to block tri ..."
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Cited by 13 (7 self)
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Abstract. For the nonHermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skewHermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skewHermitian splitting iteration methods. 1.
Analysis of a Preconditioned Iterative Method for the ConvectionDiffusion Equation
, 2003
"... A preconditioner for the convectiondiffusion equation based on the diffusion part is considered. The algorithm is compared with a strategy based on a twostep iterative method (HSS) for the solution of nonsymmetric linear systems whose real part is coercive and with other preconditioned algorithms ..."
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Cited by 5 (2 self)
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A preconditioner for the convectiondiffusion equation based on the diffusion part is considered. The algorithm is compared with a strategy based on a twostep iterative method (HSS) for the solution of nonsymmetric linear systems whose real part is coercive and with other preconditioned algorithms based on incomplete factorization techniques. The analysis of the distribution of the eigenvalues of the preconditioned matrix and of the convergence of iterations are provided as well, showing that the preconditioned iterations do not depend on the mesh, provided that it is fine enough. Moreover, it is shown that the overall cost of the algorithm is O(n), where n is the size of the underlying matrices.
Optimal Parameter in Hermitian and SkewHermitian Splitting Method for Certain Twobytwo Block Matrices
, 2005
"... The optimal parameter of the Hermitian/skewHermitian splitting (HSS) iteration method for a real 2by2 linear system is obtained. The result is used to determine the optimal parameters for linear systems associated with certain 2by2 block matrices, and to estimate the optimal parameters of the H ..."
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Cited by 5 (0 self)
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The optimal parameter of the Hermitian/skewHermitian splitting (HSS) iteration method for a real 2by2 linear system is obtained. The result is used to determine the optimal parameters for linear systems associated with certain 2by2 block matrices, and to estimate the optimal parameters of the HSS iteration method for linear systems with nbyn real coefficient matrices. Numerical examples are given to illustrate the results.
Superlinear Convergence of a Preconditioned Iterative Method for the ConvectionDiffusion Equation
, 2004
"... The convergence features of a preconditioned algorithm for the convectiondiffusion equation based on its diffusion part are considered. An analysis of the distribution of the eigenvalues of the preconditioned matrix and of the fundamental parameters of convergence are provided, showing the existenc ..."
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Cited by 1 (1 self)
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The convergence features of a preconditioned algorithm for the convectiondiffusion equation based on its diffusion part are considered. An analysis of the distribution of the eigenvalues of the preconditioned matrix and of the fundamental parameters of convergence are provided, showing the existence of a proper cluster of eigenvalues and the superlinear behavior of preconditioned iterations. The structure of the cluster is not influenced by the discretization but the number of outliers increases at most proportionally to &nu; as the viscosity parameter &nu; is increased. The overall cost of the algorithm is O(n), where n ist the size of the underlying matrices.
A Note on Multigrid Methods for (Multilevel) Structuredplusbanded Uniformly Bounded Hermitian Positive Definite Linear Systems
, 2008
"... In the past few years a lot of attention has been paid in the multigrid solution of multilevel structured (Toeplitz, circulants, Hartley, sine (τ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In the presen ..."
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In the past few years a lot of attention has been paid in the multigrid solution of multilevel structured (Toeplitz, circulants, Hartley, sine (τ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In the present paper we provide some theoretical results on the optimality of an existing multigrid procedure, when applied to a properly related algebraic problem. In particular, we propose a modification of previously devised multigrid procedures in order to handle Hermitian positive definite structuredplusbanded uniformly bounded linear systems, arising when an indefinite, and not necessarily structured, banded part is added to the original coefficient matrix. In this context we prove the TwoGrid method optimality. In such a way, several linear systems arising from the approximation of integrodifferential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to TwoGrid and multigrid procedures.
AND
, 2011
"... doi:10.1093/imanum/drs001 Preconditioned MHSS iteration methods for a class of block twobytwo linear systems with applications to distributed control problems ..."
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doi:10.1093/imanum/drs001 Preconditioned MHSS iteration methods for a class of block twobytwo linear systems with applications to distributed control problems
symmetric linear systems
"... Modified HSS iteration methods for a class of complex ..."
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