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15
Block preconditioning of realvalued iterative algorithms for complex linear systems
, 2008
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A Relaxed Dimensional Factorization Preconditioner for the Incompressible NavierStokes Equations
, 2010
"... In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. Nu ..."
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Cited by 9 (3 self)
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In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.
On an augmented Lagrangianbased preconditioning of Oseen type problems
"... The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized NavierStokes equations (Oseen’s problem). We utilize the socalled augmented Lagrangian approach, where the original linear system of equations is ..."
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Cited by 7 (6 self)
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The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized NavierStokes equations (Oseen’s problem). We utilize the socalled augmented Lagrangian approach, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method. The matrices in the linear systems, arising after the discretization of Oseen’s problem, are of twobytwo block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the blockpreconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods. The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an elementbyelement fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy. We also discuss possible approaches to solve the modified pivot matrix block. Keywords: NavierStokes equations, saddle point systems, augmented Lagrangian, finite elements, approximation of mass matrixiterative methods, preconditioning 1
On preconditioned MHSS iteration methods for complex . . .
, 2011
"... We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the convergence of the preconditioned MHSS (PMHSS) iteration method and discuss the spectral properties of the PMH ..."
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Cited by 7 (4 self)
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We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the convergence of the preconditioned MHSS (PMHSS) iteration method and discuss the spectral properties of the PMHSSpreconditioned matrix. Numerical implementations show that the resulting PMHSS preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as GMRES and its
Optimization of the parameterized Uzawa preconditioners for saddle point matrices
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2009
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On OrderReducible Sinc Discretizations and BlockDiagonal Preconditioning Methods for Linear ThirdOrder Ordinary Differential Equations ∗
, 2012
"... By introducing a variable substitution we transform the twopoint boundary value problem of a thirdorder ordinary differential equation into a system of two secondorder ordinary differential equations. We discretize this orderreduced system of ordinary differential equations by both sinccollocat ..."
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Cited by 1 (1 self)
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By introducing a variable substitution we transform the twopoint boundary value problem of a thirdorder ordinary differential equation into a system of two secondorder ordinary differential equations. We discretize this orderreduced system of ordinary differential equations by both sinccollocation and sincGalerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the orderreduced system of ordinary differential equations. The coefficient matrix of the linear system is of block twobytwo structure and each of its blocks is a combination of Toeplitz
Preconditioned . . . for a Class of Block TwobyTwo Linear Systems with Applications to Distributed Control Problems
, 2011
"... We construct a preconditioned MHSS (PMHSS) iteration scheme for solving and preconditioning a class of block twobytwo linear systems arising from the Galerkin finiteelement discretizations of a class of distributed control problems. The convergence theory of this class of PMHSS iteration methods ..."
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We construct a preconditioned MHSS (PMHSS) iteration scheme for solving and preconditioning a class of block twobytwo linear systems arising from the Galerkin finiteelement discretizations of a class of distributed control problems. The convergence theory of this class of PMHSS iteration methods is established and the spectral properties of the PMHSSpreconditioned matrix are analyzed. Numerical experiments show that the PMHSS preconditioners can be quite competitive when used to precondition Krylov subspace iteration methods such as GMRES.
of Complex Symmetric Linear Systems ∗
, 2009
"... In this paper, we introduce and analyze a modification of the Hermitian and skewHermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skewHermitian splitting (MHSS) iteration method is unconditionally convergent. ..."
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In this paper, we introduce and analyze a modification of the Hermitian and skewHermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skewHermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method.
Preconditioned Iterative Methods for Algebraic Systems from Multiplicative HalfQuadratic Regularization Image Restorations
, 2009
"... Image restoration is often solved by minimizing an energy function consisting of a datafidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edgepreserving regularization ..."
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Image restoration is often solved by minimizing an energy function consisting of a datafidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edgepreserving regularization functions, i.e., multiplicative halfquadratic regularizations, and we
A Relaxed Dimensional Factorization Preconditioner for the Incompressible NavierStokes Equations
"... In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. ..."
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In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity. Key words: saddle point problem, Navier–Stokes equations, Oseen problem, Krylov subspace method, dimensional splitting, dimensional factorization 1.