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35
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 322 (25 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. SCI. COMPUT
, 2006
"... We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel mult ..."
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Cited by 68 (19 self)
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We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2P0 and isoP2P1 finite elements in support of our conclusions. We also show results of a comparison with two stateoftheart preconditioners, showing the competitiveness of our approach.
A preconditioner for generalized saddle point problems
 SIAM J. Matrix Anal. Appl
, 2004
"... 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/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matri ..."
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Cited by 44 (23 self)
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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/ skewsymmetric 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
Domain decomposition preconditioners for linear–quadratic elliptic optimal control problems
, 2004
"... We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic ell ..."
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Cited by 19 (4 self)
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We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size.
Iterative Linear Algebra for Constrained Optimization
, 2005
"... Each step of an interior point method for nonlinear optimization requires the solution of a symmetric indefinite linear system known as a KKT system, or more generally, a saddle point problem. As the problem size increases, direct methods become prohibitively expensive to use for solving these probl ..."
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Cited by 13 (4 self)
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Each step of an interior point method for nonlinear optimization requires the solution of a symmetric indefinite linear system known as a KKT system, or more generally, a saddle point problem. As the problem size increases, direct methods become prohibitively expensive to use for solving these problems; this leads to iterative solvers being the only viable alternative. In this thesis we consider iterative methods for solving saddle point systems and show that a projected preconditioned conjugate gradient method can be applied to these indefinite systems. Such a method requires the use of a specific class of preconditioners, (extended) constraint preconditioners, which exactly replicate some parts of the saddle point system that we wish to solve. The standard method for using constraint preconditioners, at least in the optimization community, has been to choose the constraint
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.
AN EFFICIENT NUMERICAL METHOD FOR THE SOLUTION OF THE L2 OPTIMAL MASS TRANSFER PROBLEM
, 2010
"... In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannen ..."
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Cited by 12 (0 self)
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In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61–97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data.
A general approach to analyse preconditioners for twobytwo block matrices
"... Twobytwo block matrices arise in various applications, such as in domain decomposition methods or, more generally, when solving boundary value problems discretized by finite elements from the separation of the node set of the mesh into ’fine’ and ’coarse’ nodes. Matrices with such a structure, in ..."
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Cited by 11 (6 self)
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Twobytwo block matrices arise in various applications, such as in domain decomposition methods or, more generally, when solving boundary value problems discretized by finite elements from the separation of the node set of the mesh into ’fine’ and ’coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimization problems. A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in twobytwo block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve elementbyelement approximations and/or geometric or algebraic multigrid/multilevel